mes47759_ch10_625-641 09/27/2007 06:18 Page 625 pinnacle 201:MHIA038:mhmes2:mes2ch10: CHAPTER 10 Radicals and Rational Exponents 10.1 Finding Root...
Author: Fay Johnson
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CHAPTER

10

10.1 Finding Roots •••

Algebra at Work: Forensics Forensic scientists use mathematics in many ways to help them analyze evidence and solve crimes. To help him reconstruct an accident scene, Keith can use this formula containing a radical to estimate the minimum speed of a vehicle when the accident occurred: S  130fd where f  the drag factor, based on the type of road surface d  the length of the skid, in feet S  the speed of the vehicle in miles per hour

10.2 Rational Exponents ••• 10.3 Simplifying Expressions Containing Square Roots ••• 10.4 Simplifying Expressions Containing Higher Roots ••• 10.5 Adding and Subtracting Radicals •••

Keith is investigating an accident

in a residential neighborhood

where the speed limit is 25 mph.

The car involved in the accident left skid marks 60 ft long. Tests showed that the drag factor of the asphalt road was 0.80. Was the driver speeding at the time of the accident?

Substitute the values into the equation and evaluate it to determine the minimum speed of the vehicle at the time of the accident: S  230fd S  230(0.80)(60) S  21440  38 mph The driver was going at least 38 mph when the accident occurred. This is well over the speed limit of 25 mph. We will learn how to simplify radicals in this chapter as well as how to work with equations like the one given here. 625

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Chapter 10 Radicals and Rational Exponents

Section 10.1 Finding Roots Objectives 1. Find the Square Root of a Rational Number 2. Approximate the Square Root of a Whole Number 3. Find the Higher Roots of Rational Numbers

In Section 1.2 we introduced the idea of exponents as representing repeated multiplication. For example, 32 means 3  3, so 32  9. 24 means 2  2  2  2, so 24  16. In this chapter we will study the opposite procedure, finding roots of numbers.

1. Find the Square Root of a Rational Number Example 1 Find all square roots of 25.

In-Class Example 1 Find all square roots of 81. answer: 9, 9

Solution To find a square root of 25 ask yourself, “What number do I square to get 25?” Or, “What number multiplied by itself equals 25?” One number is 5 since 52  25. Another number is 5 since (5) 2  25. 5 is a square root of 25. 5 is a square root of 25.

You Try 1 Find all square roots of 64.

The 1

symbol represents the positive square root of a number. For example, 125  5

125  5 but 125  5. The 1

symbol represents only the positive square root.

To find the negative square root of a number we must put a  in front of the 1 . For example, 125  5. Next we will define some terms associated with the 1

symbol.

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Section 10.1 Finding Roots

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The entire expression, 125, is called a radical.

Example 2

a)

In-Class Example 2 Find each square root. a) 116 b) 19 36 100 c) d)  B 49 B 9 answer: a) 4 10 d)  3

b) 3

Find each square root.

6 c) 7

2100

b) 216

c)

4 B 25

81 d)  B 49

Solution a) 1100  10 since (10) 2  100. b) 116 means 1  116. Therefore, 116  1  116  1  (4)  4 c) Since 14  2 and 125  5,

4 2  . B 25 5

81 81 d)  means 1  . Therefore, B 49 B 49 81 81 9 9   1   1  a b   B 49 B 49 7 7

You Try 2 Find each square root. a) 1144

Example 3 In-Class Example 3 Find 164. answer:There is no such real number.

b)

25 B 36

1 c)  B 64

Find 19.

Solution Recall that to find 19 you can ask yourself, “What number do I square to get 9?” or “What number multiplied by itself equals 9?” There is no such real number since 32  9 and (3) 2  9. Therefore, 19 is not a real number.

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Chapter 10 Radicals and Rational Exponents

You Try 3 Find 136.

Let’s review what we know about a square root and the radicand and add a third fact.

1)

If the radicand is a perfect square, the square root is a rational number. Example: 116  4 100 10  B 49 7

2)

16 is a perfect square. 100 is a perfect square. 49

If the radicand is a negative number, the square root is not a real number. Example: 125 is not a real number.

3)

If the radicand is positive and not a perfect square, then the square root is an irrational number. Example: 113 is irrational.

(13 is not a perfect square)

The square root of such a number is a real number that is a nonrepeating, nonterminating decimal. It is important to be able to approximate such square roots because sometimes it is necessary to estimate their places on a number line or on a Cartesian coordinate system when graphing.

For the purposes of graphing, approximating a radical to the nearest tenth is sufficient. A calculator with a 1 key will give a better approximation of the radical.

2. Approximate the Square Root of a Whole Number Example 4 In-Class Example 4 Approximate 117 to the nearest tenth and plot it on a number line. answer: 4.1 17 1

0

1

2

3

4

5

6

Approximate 113 to the nearest tenth and plot it on a number line.

Solution What is the largest perfect square that is less than 13? 9 What is the smallest perfect square that is greater than 13? 16 Since 13 is between 9 and 16 (9  13  16), it is true that 113 is between 19 and 116. ( 19  113  116) 19  3 113  ? 116  4

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Section 10.1 Finding Roots

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113 must be between 3 and 4. Numerically, 13 is closer to 16 than it is to 9. So, 113 will be closer to 116 than to 19. Check to see if 3.6 is a good approximation of 113. ( means approximately equal to.) If 113  3.6, then (3.6) 2  13 (3.6) 2  (3.6)  (3.6)  12.96 Is 3.7 a better approximation of 113? If 113  3.7, then (3.7) 2  13 (3.7) 2  (3.7)  (3.7)  13.69 3.6 is a better approximation of 113. 113  3.6 √13 0

1

2

3

4

5

6

A calculator evaluates 113 as 3.6055513. Remember that this is only an approximation.

You Try 4 Approximate 129 to the nearest tenth and plot it on a number line.

3. Find the Higher Roots of Rational Numbers We saw in Example 2a) that 1100  10 since (10) 2  100. Finding a 1 is the 3 opposite of squaring a number. Similarly, we can find higher roots of numbers like 1 a 4 5 (read as “the cube root of a”), 1 a (read as “the fourth root of a”), 1 a (the fifth root of a), etc.

Example 5 In-Class Example 5 Find each root. 4 3 3 a) 1 16 b) 1 27 c) 1 64 answer: a) 2 b) 3 c) 4

Find each root. 3 a) 1 125

b)

4 1 81

c)

5 1 32

Solution 3 a) To find 1 125 (read as “the cube root of 125”) ask yourself, “What number do I cube to get 125?” That number is 5. 3 1 125  5 since 53  125

Finding the cube root of a number is the opposite of cubing a number. 4 b) To find 181 (read as “the fourth root of 81”) ask yourself, “What number do I raise to the fourth power to get 81?” That number is 3. 4 1 81  3 since 34  81

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Chapter 10 Radicals and Rational Exponents

Finding the fourth root of a number is the opposite of raising a number to the fourth power. 5 c) To find 1 32 (read as “the fifth root of 32”) ask yourself, “What number do I raise to the fifth power to get 32?” That number is 2. 5 1 32  2 since 25  32

Finding the fifth root of a number is the opposite of raising a number to the fifth power.

You Try 5 Find each root. 4

3

a) 1 16

b) 1 27

We can use a general notation for writing roots of numbers.

n

n

The 1 a is read as “the nth root of a.” If 1 a  b, then bn  a. n is the index of the radical.

2 When finding square roots we do not write 1 a. The square root of a is written as 1a, and the index is understood to be 2.

In Section 1.2 we first presented the powers of numbers that students are expected to know. (22  4, 23  8, etc.) Use of these powers was first necessary in the study of the rules of exponents in Chapter 2. Knowing these powers is necessary for finding roots as well, so the student can refer to p. ••• to review this list of powers. While it is true that the square root of a negative number is not a real number, sometimes it is possible to find the higher root of a negative number.

Example 6 In-Class Example 6 Find each root, if possible 3 4 a) 1 27 b) 1 81 4 3 c) 181 d) 1 125 answer: a) 3 b) 3 c) Not a real number d) 5

Find each root, if possible. 3 5 a) 1 b) 1 64 32

4 c) 1 16

d)

4 1 16

Solution 3 a) To find 1 64 ask yourself, “What number do I cube to get 64?” That number is 4. 3 1 64  4 since (4) 3  64

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Section 10.1 Finding Roots

631

5 b) To find 1 32 ask yourself, “What number do I raise to the fifth power to get 32?” That number is 2. 5 1 32  2 since (2) 5  32 4 4 c) 116 means 1  1 16. Therefore, 4 4 1 16  1  1 16  1  2  2 4 d) To find 1 16 ask yourself, “What number do I raise to the fourth power to get 16?” There is no such real number since 24  16 and (2) 4  16. 4

1 16 is not a real number. We can summarize what we have seen in example 6 as follows:

1)

The odd root of a negative number is a negative number.

2)

The even root of a negative number is not a real number.

You Try 6 Find each root, if possible. 6 a) 1 64

3 b) 1 125

4 c) 1 81

Answers to You Try Exercises 1) 8 and 8

2) a) 12

5 6

b)

c) 

1 8

3) not a real number

√29

4) 5.4

0

1

2

3

6) a) not a real number

10.1 Exercises

4

5

6

b) 5

5) a) 2 c) 3

 Practice Problems

b) 3

 NetTutor  Self-Test

 e-Professors

 Videos

Objective 1

Decide if each statement is true or false. If it is false, explain why. 1) 1121  11 and 11

2) 181  9

True

3) The cube root of a negative number is a negative number. True

4) The square root of a negative number is a negative False; the square root of a negative number. number is not a real number.

5) The even root of a negative number is a negative False; the even root of a negative number. number is not a real number.

6) The odd root of a negative number is a negative number. True

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Chapter 10 Radicals and Rational Exponents

39) 117

40) 169

12 and 12

41) 15

42) 135

43) 161

44) 18

Find all square roots of each number.

video

7) 49

7 and 7

9) 1

1 and 1

10) 81

9 and 9

11) 400

20 and 20

12) 900

30 and 30

13) 2500

50 and 50

14) 4900 70 and 70

4 9

2 2 and  3 3

16)

1 81

1 1 and  9 9

18)

15) 17)

8) 144

Objectives 2 and 4 3 45) Explain how to find 1 64.

36 25

6 6 and  5 5

4 46) Explain how to find 1 16.

1 16

1 1 and  4 4

48) Does 1 8  2? Why or why not?

4 47) Does 1 81  3? Why or why not? 3

Find each root, if possible.

Find each square root, if possible. 19) 149 21) 11 23) 1169 25) 14

20) 1144

7

22) 181

1 13 not real

24) 136 26) 1100

3

49) 1 8

12

2

3

51) 1 125

9

5

3

53) 1 1

6

1

4

55) 1 81

not real

3

4

27) 29) video

81 B 25 49 B 64

31) 136 1 33)  B 121

9 5

28)

16 B 169

30)

121 B 4

11 2

6

32) 164

8

1 34)  B 100



1 11

59) 1 16

video

35) 111

36) 12

37) 146

38) 122

61) 1 32

2

63) 1 27

3

6

65) 1 64

1

3

52) 1 27

3

3

54) 1 8

2

4

56) 1 16

2

4

58) 1 81

not real

4

1

6

2

60) 1 1 62) 1 64 3

64) 1 1000 4

not real

66) 116

3 8 A 125

2 5

68)

69) 160  11

7

70) 1100  21

67) video

Approximate each square root to the nearest tenth and plot it on a number line.

2

5

3

1 10

Objective 2

not real

4

video

7 8



57) 1 1

4 13

3

50) 1 1

3

10 not real

4 81 A 16

3 2

3

11

71) 1 100  25

5

72) 1 9  36

3

73) 11  9

not real

74) 125  36

not real

13

76) 23  4

5

75) 25  12 2

2

2

2

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Section 10.2 Rational Exponents

633

Section 10.2 Rational Exponents Objectives 1. Define and Evaluate Expressions of the Form a1n 2.

3.

4.

5.

Define and Evaluate Expressions of the Form amn Define and Evaluate Expressions of the Form amn Combine the Rules of Exponents to Simplify Expressions Convert a Radical Expression to Exponential Form and Simplify

In-Class Example 1 Write in radical form and evaluate. a) 912 b) 2713 c) 3215 answer: a) 3 b) 3 c) 2

1. Define and Evaluate Expressions of the Form a1n In this Section, we will explain the relationship between radicals and rational (fractional) exponents. Sometimes, converting between these two forms makes it easier to simplify expressions.

If a is a nonnegative number and n is a positive integer, then a1  n  1 a n

(The denominator of the fractional exponent is the index of the radical.)

Example 1 Write in radical form and evaluate. a) 813 b) 4912 c) 8114

Solution a) The denominator of the fractional exponent is the index of the radical. Therefore, 3 813  1 8  2.

b) The denominator in the exponent of 4912 is 2, so the index on the radical is 2, meaning square root. 491 2  249  7. 4 c) 8114  1 81  3

You Try 1 Write in radical form and evaluate. a) 1614

b) 1211 2

2. Define and Evaluate Expressions of the Form a mn We can add another relationship between rational exponents and radicals.

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Chapter 10 Radicals and Rational Exponents

If a is a nonnegative number and m and n are integers such that n is positive, amn  (a1n ) m  ( 1a) m n

(The denominator of the fractional exponent is the index of the radical, and the numerator is the power to which we raise the radical expression.)

Alternatively, we can think of amn in this way: amn  (am ) 1n  1am. n

Example 2 In-Class Example 2 Write in radical form and evaluate. a) 823 b) 1632 answer: a) 4 b) 64

Write in radical form and evaluate. a) 2532 b) 6423

Solution a) According to the definition, the denominator of the fractional exponent is the index of the radical, and the numerator is the power to which we raise the radical expression. 2532  (251 2 ) 3  ( 125) 3  53  125

Use the definition to rewrite the exponent. Rewrite as a radical. 225  5

b) To evaluate 6423, first evaluate 6423, then take the negative of that result. 6423  (6423 )  (6413 ) 2 3  ( 1 64) 2 2  (4)  16

Use the definition to rewrite the exponent. Rewrite as a radical. 3 1 64  4

You Try 2 Write in radical form and evaluate. a) 3225

b) 10032

3. Define and Evaluate Expressions of the Form a mn Recall the definition of a negative exponent from Section 2.4. If n is any integer and a  0, then

1 n 1 an  a b  n . a a

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Section 10.2 Rational Exponents

635

That is, to rewrite the expression with a positive exponent, take the reciprocal of the base. For example, 1 4 1 24  a b  2 16 We can extend this to rational exponents. If a is a positive number and m and n are integers such that n is positive, 1 mn 1 amn  a b  mn . a a (To rewrite the expression with a positive exponent, take the reciprocal of the base.)

Example 3 In-Class Example 3 Rewrite with a positive exponent and evaluate. a) 93 2 b) 8134 8 43 c) a b 27 1 1 answer: a) b) 27 3 81 c) 16

Rewrite with a positive exponent and evaluate. a) 3612

c) a

b) 3225

125 23 b 64

Solution a) To write 3612 with a positive exponent, take the reciprocal of the base. 1 1 2 b 36 1  B 36 1  6

36 1 2  a

1 25 b 32 1 2 a5 b A 32 1 2 a b 2 1  4

3225  a

b)

c)

a

125 23 64 23 a b b 64 125 64 2 a3 b A 125 4 2 a b 5 16  25

The reciprocal of 36 is

1 . 36

The denominator of the fractional exponent is the index of the radical.

The reciprocal of 32 is

1 . 32

The denominator of the fractional exponent is the index of the radical. 1 5 1  A 32 2

The reciprocal of

125 64 is . 64 125

The denominator of the fractional exponent is the index of the radical. 64 4  A 125 5 3

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Chapter 10 Radicals and Rational Exponents

The negative exponent does not make the expression negative!

You Try 3 Rewrite with a positive exponent and evaluate. a) 1441 2

c) a

b) 1634

8 23 b 27

We can combine the rules presented in this section with the rules of exponents we learned in Chapter 2 to simplify expressions containing numbers or variables.

4. Combine the Rules of Exponents to Simplify Expressions Example 4 In-Class Example 4 Simplify completely.The answer should contain only positive exponents a) (325 ) 6 b) 2723  2713 4133 c) 7 3 4 answer: a) 3125 b) 3 c) 16

Simplify completely. The answer should contain only positive exponents. 829 a) (615 ) 2 b) 2534  2514 c) 119 8

Solution a) (615 ) 2  625 3 1 b) 2534  2514  254  14 2

Multiply exponents. Add exponents. 3 1 Add  a b. 4 4

24

 25

2 Reduce . 4 Evaluate.

 2512 5 c)

82  9 8

119

 89  9

Subtract exponents.

 899

Subtract

2

11

2 11  . 9 9

9 Reduce  . 9

 81 1 1 a b 8 1  8

Rewrite with a positive exponent.

You Try 4 Simplify completely.The answer should contain only positive exponents. a) 4938  4918

b) (16112 ) 3

c)

725 745

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Section 10.2 Rational Exponents

Example 5 In-Class Example 5 Simplify completely. Assume the variables represent positive real numbers.The answer should contain only positive exponents. a) x14  x54 c)

h23  h34

b) a

32

q

h53

p1  6

b)

8

b

p43 q12

1 74

h

637

Simplify completely. Assume the variables represent positive real numbers. The answer should contain only positive exponents. x23 6 b) a 14 b y

a) r18  r38

Solution 1 3 a) r18  r38  r 8  8 48 r  r12

c)

n56  n13 n16

x23 6 x3 6 b) a 14 b  1 6 y y4 x4  32 y 5 1 56 13 n n n6  3 c)  16 n16 n 5 2 n6  6  16 n n36  16 n 3 1  n6  16 2 2

26

n

 n13 1  13 n

4 Reduce . 8 Multiply exponents. Multiply and reduce. Add exponents. Get a common denominator. Add exponents. Subtract exponents. 3 1 3 1 2   a b      6 6 6 6 6 Reduce. Rewrite with a positive exponent.

You Try 5 Simplify completely. Assume the variables represent positive real numbers.The answer should contain only positive exponents. a) (a3b15 ) 10

b)

t310 t710

c)

s34 s12  s54

5. Convert a Radical Expression to Exponential Form and Simplify Some radicals can be simplified by first putting them into rational exponent form and then converting them back to radicals.

Example 6 In-Class Example 6 Follow the instructions for Example 6. 10 8 a) 2 365 b) 2x6 c) ( 13) 2 4 d) 274 4 answer: a) 6 b) 2x3 c) 3 d) 7

Rewrite each radical in exponential form, then simplify. Write the answer in simplest (or radical) form. Assume all variables represent positive real numbers. a)

8 4 2 9

6 b) 2s4

c) ( 17) 2

3 d) 253

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Chapter 10 Radicals and Rational Exponents

Solution a) Since the index of the radical is the denominator of the rational exponent and the power is the numerator, we can write 8 4 2 9  948  912 3 6 4 2s  s46  s23

b)

Write with a rational exponent. 4 Reduce . 8 Evaluate. Write with a rational exponent. 4 Reduce . 6 Write in radical form.

3 2 2 s

6 4 2 s is not in simplest form because the 4 and the 6 contain a common factor of 2. 3 2 2 s is in simplest form because 2 and 3 do not have any common factors besides 1.

c)

d)

( 17) 2  (712 ) 2 1  72 2  71 7 3 3 (2 5 )  (53 ) 13 1  53 3 1 5 5

Write with a rational exponent. Multiply exponents.

Write with a rational exponent. Multiply exponents.

You Try 6 Rewrite each radical in exponential form, then simplify.Write the answer in simplest (or radical) form. Assume all variables represent positive real numbers. 6 a) 21252

10

b) 2 p4

c) ( 16) 2

3 d) 293

Example 6 d) illustrates an important and useful property of radicals.

If a is a nonnegative real number and n is a positive integer, then n

n

( 1a) n  a and 1 an  a.

We can show, in general, why this is true by writing the expressions with rational exponents. ( 1a) n  ann  a1  a n 1 an  ann  a1  a n

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Section 10.2 Rational Exponents

Example 7 In-Class Example 7 Simplify. Assume all variables represent positive real numbers. 3 4 4 a) ( 2 11) 3 b) 2 7 2 c) ( 2m) answer: a) 11 b) 7

c) m

Simplify. Assume the variable represents a positive real number. 3 4 4 a) ( 2 b) 2 c) 1 2x2 2 4) 3 5

Solution 3 a) Since the index of the radical is the same as the exponent, ( 1 4) 3  4. 4 4 b) 25  5 c) ( 1x) 2  x

You Try 7 Simplify. Assume all variables represent positive real numbers. a) ( 210) 2

3 3 b) 2 7

c) 2t2

Using Technology Now that we have discussed the relationship between rational exponents and radicals, let’s learn how to evaluate these expressions on a graphing calculator. Example 1 Evaluate 912. Using the properties in this chapter, we get 91 2  19  3 We will look at two ways to correctly enter the expression into the calculator. 1.

2.

The ^ key indicates that an exponent follows.To enter and evaluate 912, press this sequence of keys: 9 ^ ( 1  2 ) ENTER .The result is 3. You must use parentheses! If you forget the parentheses and enter 9 ^ 1  2 , the result will be 4.5 because the calculator is following the order of operations. To enter 912 as 19, press this sequence of keys: 2nd x2 9 ) ENTER .Again, the result is 3. Notice that the calculator inserted the left parenthesis automatically.You need to close the parentheses so that the calculator knows where the square root ends.

Example 2 Evaluate 813. Using the properties in this chapter, we get 3 813  1 8  2.

1.

To evaluate 813 on the calculator using the ^ key, press 8 ^ ( 1  3 ) ENTER . The result is 2.

639

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Chapter 10 Radicals and Rational Exponents

3 Roots higher than square roots can be found by pressing the MATH key.To enter 1 8, 3 press MATH and use the down arrow to highlight 4: 1 (. Press ENTER .Then press 8 ) ENTER , and you will see that the result is 2.

2.

Evaluate each expression using the properties learned in this chapter, then verify your answer using the graphing calculator. 1.

4912

5.

1213

2.

14412

6.

6413

3.

8114

7.

1632

4.

3215

8.

2743

Answers to You Try Exercises 1) a) 2

b) 11

5) a) a30b2

b)

b) 1000

2) a) 4 1 2/5

t

c) s32

6) a) 5

1 12

3) a)

b)

1 8

5 b) 2p2 c) 6

c)

9 4

d) 9

4) a) 7

b) 2

7) a) 10

b) 7

c)

1 725

c) t

Answers to Technology Exercises 1. 7

2. 12

3. 3

4. 2

5. 5

7. 64

 Practice Problems

10.2 Exercises

6. 4

8. 81

 NetTutor  Self-Test

 e-Professors

 Videos

Objective 1 12

1) Explain how to write 25

2) Explain how to write 113 in radical form. video

Write in radical form and evaluate. 3) 912

3 13

4) 6412

8

13

5) 1000

10

6) 27

3

7) 3215

2

8) 8114

3

12

5

10) 641/6 12

11) a

4 b 121

2 11

4 12) a b 9

13) a

125 13 b 64

5 4

14) a

15) a

36 12 b 169



6 13

16 14 b 81

16) a

100 12 b 9

2 2 3 2 3 

10 3

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Section 10.2 Rational Exponents

Objective 2 3/4

17) Explain how to write 16

59)

3/2

18) Explain how to write 100

61)

Write in radical form and evaluate. video

19) 843

16

20) 8134

27

21) 12523

25

22) 3235

8

56

23

32

24) 1000

100

25) 2743

81

26) 3632

216

16 34 27) a b 81

8 27

64 23 b 28) a 125

16 25

1000 23 b 29) a 27

100  9

8 43 30) a b 27

16  81

35) 100013 1 14 37) a b 81 39) a video

1 13 b 64

41) 6456 43) 12523 25 32 b 4

45) a 47) a

23

64 b 125

34) 10012 36) 2713

3

1 15 38) a b 32

4

40) a

1 32 1 25

42) 8134

8 125

46) a

25 16

1 13 b 125

44) 6423

48) a

9 32 b 100 34

81 b 16

j

71)

20c23 56

72c

73) (q

45 10

)

79) (81u83v4 ) 34 f 67

x53

2

5

87)

1 27 1 16 1000 27

89) a

6

y12  y13

1

y56

y23

a4b3 25 b 32a2b4

video

6 91) 2493

7

57)

443 532 5

92

1 125

58)

234 3235 32

75

1 16

2

2

3

3

4

4

99) ( 1 12) 101) ( 1 15) 3

32

32x5y2

84) a

88)

a125 4b

25

34

16c8

8b114

b 113

c6

b

t32

4

b u14

t6u

t5 t12  t34

90) a

25a6b16

t154

16c8d3 32 b c4d5

64 c18d3

Rewrite each radical in exponential form, then simplify. Write the answer in simplest (or radical) form. Assume all variables represent positive real numbers.

3

2234

1

Objective 5

51) (914 ) 2

56)

5w110

80) (64x6y125 ) 56

86) a

x10w9

97) 232

16

f 27g59 3

b w3  2

8

4103

24

25

n6  7 43 52 78) (r ) r103 27u2v3

13

85) a

613

10w

1

81) (32r13s49 ) 35 8r15s415 82) (125a9b14 ) 23 b 27g53

49

48w310

76) (n27 ) 3

x23 z215

83) a

25

h

z

34

74) (t38 ) 16 t6

1

77) (z15 ) 23 video

72)

q

75) (x29 ) 3

1 10 1 3

8 27

h712 68) (3x13 )(8x49 ) 24x19 x16 1 70) 56 x23 x

8

49) 223  273

55)

4

465  435

66) h

18c32

95) ( 15)

54) 643  653

1 75

j310

5

93) 21000

84  5

425

16

a

a49

Simplify completely. The answer should contain only positive exponents.

53) 875  835

1000

64) z16  z56

19

6

52) 1723 2 3

1052  1032 104

k52 1

72v118

a59

Objective 4

50) 534  554

62)

67) (9v58 )(8v34 )

Rewrite with a positive exponent and evaluate. 33) 4912

713

310

65) j

69)

1 32 1 23 b a b 32) a 100 100 1 7 1 10

60)

729

35

Decide whether each statement is true or false. Explain your answer.  9

749  719

63) k74  k34

Objective 3

31) 81

6

6

6

52

Simplify completely. Assume the variables represent positive real numbers. The answer should contain only positive exponents.

23) 64

12

61 12

641

9 92) 283 4

2

10

94) 281

5

96) ( 114)

3

98) 272

12 15

2

9 2

7

3

100) ( 1 10) 4

102) ( 19) 4 8

14

4

3

10 9

103) 2x

x4

104) 2t

t2

5 105) 2k 2

3 1 k

9 106) 2w6

3 2 w2

4 107) 2z2

1z

8 108) 2m4

1m

109) 2d

2

110) 2s

s3

12

4

d

6

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642

Chapter 10 Radicals and Rational Exponents

Section 10.3 Simplifying Expressions Containing Square Roots Objectives 1. Multiply Square Roots 2. Simplify the Square Root of a Whole Number 3. Use the Quotient Rule for Square Roots 4. Simplify Square Root Expressions Containing Variables with Even Exponents 5. Simplify Square Root Expressions Containing Variables with Odd Exponents 6. Multiply, Divide, and Simplify Expressions Containing Square Roots

Example 1 In-Class Example 1 Multiply. Assume the variable represents a nonnegative real number. a) 17  13 b) 15  1g answer: a) 121 b) 15g

In this section, we will introduce rules for finding the product and quotient of square roots as well as for simplifying expressions containing square roots.

1. Multiply Square Roots We begin with an example that we can evaluate using the order of operations: Evaluate 14  19. 14  19  2  3 6 Since both roots are square roots, however, we can also evaluate the product this way: 14  19  14  9  136 6 We obtain the same result. This leads us to the product rule for multiplying expressions containing square roots. Definition

Product Rule for Square Roots Let a and b be nonnegative real numbers. Then, 1a  1b  1a  b

Multiply. Assume the variable represents a nonnegative real number. a) 15  12 b) 13  1x

Solution a) 15  12  15  2  110 b) 13  1x  13  x  13x

We can multiply radicals this way only if the indices are the same.We will see later how to multiply 3 radicals with different indices such as 15  1 t.

You Try 1 Multiply. Assume the variable represents a nonnegative real number. a) 16  15

b) 1101r

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Section 10.3 Simplifying Expressions Containing Square Roots

643

2. Simplify the Square Root of a Whole Number Knowing how to simplify radicals is very important in the study of algebra. We begin by discussing how to simplify expressions containing square roots. How do we know when a square root is simplified?

When Is a Square Root Simplified? An expression containing a square root is simplified when all of the following conditions are met: 1)

The radicand does not contain any factors (other than 1) that are perfect squares.

2)

The radicand does not contain any fractions.

3)

There are no radicals in the denominator of a fraction.

Note: Condition 1) implies that the radical cannot contain variables with exponents greater than or equal to 2, the index of the square root.

We will discuss higher roots in Section 10.4. To simplify expressions containing square roots we reverse the process of multiplying. That is, we use the property that says 1a  b  1a  1b where a or b are a perfect squares.

Example 2 In-Class Example 2 Simplify completely. a) 145 b) 172 c) 112 d) 115 answer: a) 315 b) 612 c) 213 d) 115

Simplify completely. a) 118 b) 1500

c)

121

d) 148

Solution a) The radical 118 is not in simplest form since 18 contains a factor (other than 1) that is a perfect square. Think of two numbers that multiply to 18 so that at least one of the numbers is a perfect square. 18  9  2 (While it is true that 18  6  3, neither 6 nor 3 is a perfect square.) Rewrite 118: 118  19  2  19  12  3 12

9 is a perfect square. Product rule 19  3

3 12 is completely simplified because 2 does not have any factors that are perfect squares. b) Does 500 have a factor that is a perfect square? Yes! 500  100  5. To simplify 1500, rewrite it as 1500  1100  5  1100  15  10 15

100 is a perfect square. Product rule 1100  10

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Chapter 10 Radicals and Rational Exponents

10 15 is completely simplified because 5 does not have any factors that are perfect squares. c) To simplify 121, think of two numbers that multiply to 21 so that at least one of the numbers is a perfect square. 21  3  7 21  1  21

Neither 3 nor 7 is a perfect square. While 1 is a perfect square, this will not help us simplify 121.

121 is in simplest form. d) There are two good ways to simplify 148. We will look at both of them. i) Two numbers that multiply to 48 are 16 and 3 with 16 being a perfect square. We can write 148  116  3  116  13  4 13

16 is a perfect square. Product rule 116  4

4 13 is completely simplified because 3 does not have any factors that are perfect squares. ii) We can also think of 48 as 4  12 since 4 is a perfect square. We can write 148  14  12  14  112  2 112

4 is a perfect square. Product rule 14  2

Therefore, 148  2 112. Is 112 in simplest form? No, because 12  4  3 and 4 is a perfect square. We must continue to simplify. 148  2 112  2 14  3  2 14  13  2  2  13  4 13

4 is a perfect square. Product rule 14  2 Multiply 2  2.

4 13 is completely simplified because 3 does not have any factors that are perfect squares. You can see in Example 2d) that using either 148  116  3 or 148  14  12 leads us to the same result. Furthermore, this example illustrates that a radical is not always completely simplified after just one iteration of the simplification process. It is necessary to always examine the radical to determine whether or not it can be simplified more.

After simplifying a radical, look at the result and ask yourself, “Is the radical in simplest form?” If it is not, simplify again. Asking yourself this question will help you to be sure that the radical is completely simplified.

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Section 10.3 Simplifying Expressions Containing Square Roots

645

You Try 2 Simplify completely. a) 128

b) 175

c) 172

3. Use the Quotient Rule for Square Roots We can simplify a quotient like

136 in one of two ways: 19

136 6  2 3 19

136 36  A 9 19  14 2

or

Either way we obtain the same result. This leads us to the quotient rule for dividing expressions containing square roots. Definition

Quotient Rule for Square Roots Let a and b be nonnegative real numbers such that b  0. Then, a 1a  Ab 1b

Example 3 In-Class Example 3 Simplify completely. 8 25 a) b) A 49 A5 1120 5 c) d) A 36 110 212 answer: a) b) 15 7 15 c) 213 d) 6

Simplify completely. 9 200 a) b) A 49 A 2

172 16

c)

d)

5 A 81

Solution a) Since 9 and 49 are each perfect squares, simplify

9 by finding the square A 49

root of each separately. 9 19  A 49 149 3  7

Quotient rule 19  3 and 149  7

b) Neither 200 nor 2 is a perfect square, but if we simplify the fraction 100, which is a perfect square. 200  1100 A 2  10

Simplify

200 . 2

200 we get 2

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646

Chapter 10 Radicals and Rational Exponents

172 using two different methods. 16 Begin by applying the quotient rule to obtain a fraction under one radical and simplify the fraction.

c) We can simplify i)

172 72  A6 16  112

Quotient rule 72 . 6 4 is a perfect square. Product rule 14  2 Simplify

 14  3  14  13  2 13

ii) We can apply the product rule to rewrite 172 then simplify the fraction. 172 16  112  16 16 1 16  112  16  112  14  3  14  13  213

Product rule Divide out the common factor. Simplify. 4 is a perfect square. Product rule 14  2

Either method will produce the same result. 5 d) The fraction does not reduce and 81 is a perfect square. Begin by applying 81 the quotient rule. 5 15  A 81 181 15  9

Quotient rule 181  9

You Try 3 Simplify completely. a)

100 A 169

b)

27 A3

c)

1250 15

d)

11 A 36

4. Simplify Square Root Expressions Containing Variables with Even Exponents Recall that one condition that an expression containing a square root must meet to be simplified is that the radicand cannot contain any factors (other than 1) that are perfect squares.

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Section 10.3 Simplifying Expressions Containing Square Roots

647

For a radicand containing variables this means that the radical is simplified if the power on the variable is less than 2. For example, 2r6 is not in simplified form. If r represents a nonnegative real number, then we can use rational exponents to simplify 2r6. 2r6  (r6 ) 12  r6 2  r62  r3 1

Multiplying 6 

1 is the same as dividing 6 by 2. We can generalize this with the following 2

statement:

If a is a nonnegative real number and m is an integer, then 2am  am2

We can combine this property with the product and quotient rules to simplify radical expressions.

Example 4 In-Class Example 4 Simplify completely. Assume all variables represent nonnegative real numbers. a) 2b2

b) 2100a4 27 c) 224p8 d) A g6 answer: a) b b) 10a2 313 4 c) 2p16 d) 3 g

Simplify completely. Assume all variables represent nonnegative real numbers. 32 a) 2z2 b) 249t2 c) 218b14 d) A n20

Solution a) 2z2  z22  z1  z b) 249t2  149  2t2  7  t22  7t 14 c) 218b  118  2b14  19  12  b142  3 12  b7  3b7 12 32 132 d)  A n20 2n20 116  12  n202 4 12  10 n

Product rule Evaluate. Simplify. Product rule 9 is a perfect square. Simplify. Rewrite using the commutative property. Quotient rule 16 is a perfect square. Simplify.

You Try 4 Simplify completely. Assume all variables represent positive real numbers. a) 2y10

b) 2144p16

c)

45 A w4

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Chapter 10 Radicals and Rational Exponents

5. Simplify Square Root Expressions Containing Variables with Odd Exponents How do we simplify an expression containing a square root if the power under the square root does not divide evenly by 2? We can use the product rule for radicals and fractional exponents to help us understand how to simplify such expressions.

Example 5 In-Class Example 5 Simplify completely. Assume all variables represent nonnegative real numbers a) 2m9 b) 2t13 c) 2v21 answer: a) m4 1m b) t6 1t c) v10 1v

Simplify completely. Assume all variables represent nonnegative real numbers. a) 2x7 b) 2c11 c) 2p17

Solution a) To simplify 2x7, write x7 as the product of two factors so that the exponent of one of the factors is the largest number less than 7 that is divisible by 2 (the index of the radical). 2x7  2x6  x1  2x6  1x  x62  1x  x3 1x

6 is the largest number less than 7 that is divisible by 2. Product rule Use a fractional exponent to simplify. 623

b) To simplify 2c11, write c11 as the product of two factors so that the exponent of one of the factors is the largest number less than 11 that is divisible by 2 (the index of the radical). 2c11  2c10  c1  2c10  1c  c102  1c  c5 1c

10 is the largest number less than 11 that is divisible by 2. Product rule Use a fractional exponent to simplify. 10  2  5

c) To simplify 2p17, write p17 as the product of two factors so that the exponent of one of the factors is the largest number less than 17 that is divisible by 2 (the index of the radical). 2p17  2p16  p1  2p16  1p  p162  1p  p8 1p

16 is the largest number less than 17 that is divisible by 2. Product rule Use a fractional exponent to simplify. 16  2  8

You Try 5 Simplify completely. Assume all variables represent nonnegative real numbers. a) 2m5

b) 2z19

We used the product rule to simplify each radical above. During the simplification, however, we always divided an exponent by 2. This idea of division gives us another way to simplify radical expressions. Once again let’s look at the radicals and their simplified forms in Example 5 to see how we can simplify radical expressions using division.

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Section 10.3 Simplifying Expressions Containing Square Roots

2x7  x3 2x1  x3 1x

2c11  c5 2c1  c5 1c

Index 3 S Quotient of S 2 7 radical 6 1 S Remainder

649

2p17  p8 2p1  p8 1p

Index 5 S Quotient of S 2 11 radical 10

1 S Remainder

Index 8 S Quotient of S 2 17 radical 16 1 S Remainder

To simplify a radical expression containing variables:

Example 6 In-Class Example 6 Simplify completely. Assume all variables represent nonnegative real numbers a) 2S5 b) 249v3 c) 232m19 answer: a) S2 1S b) 7v1v c) 4m9 12m

1)

Divide the exponent in the radicand by the index of the radical.

2)

The exponent on the variable outside of the radical will be the quotient of the division problem.

3)

The exponent on the variable inside of the radical will be the remainder of the division problem.

Simplify completely. Assume all variables represent nonnegative real numbers. a) 2t9 b) 216b5 c) 245y21

Solution a) To simplify 2t9, divide:

4 S Quotient 2 9 8 1 S Remainder

2t9  t4 2t1  t4 2t b) 216b5  116  2b5  4  b2 2b1  4b2 1b c) 245y21  145  2y21  19  15  y10 2y1 c Product rule

Product rule 5  2 gives a quotient of 2 and a remainder of 1. Product rule

c 21  2 gives a quotient of 10 and a remainder of 1.

 315  y10 1y  3y10  15  1y  3y10 15y

19  3 Use the commutative property to rewrite the expression. Use the product rule to write the expression with one radical.

You Try 6 Simplify completely. Assume all variables represent nonnegative real numbers. a) 2m13

b) 2100v7

c) 232a3

If a radical contains more than one variable, apply the product or quotient rule.

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Chapter 10 Radicals and Rational Exponents

Example 7

Solution a) 28a15b3  28  2a15  14  12  a7 2a1 c

In-Class Example 7 Simplify completely. Assume all variables represent nonnegative real numbers. 3f 9 a) 218m7n9 b) B g12 answer: a) 3m3n4 12mn f 4 13f b) g6

Simplify completely. Assume all variables represent nonnegative real numbers. 5r27 a) 28a15b3 b) B s8

Product rule

2b3 b1 2b1

c 15  2 gives a quotient 3  2 gives a quotient of 7 and a remainder of 1. of 1. and a remainder of 1.

 212  a7 1a  b 1b  2a7b  1a  1b  2a7b1ab

b)

 

c

650

5r27 25r27  B s8 2s8 15  2r27  s4 15r13 2r1  s4 r13  15  1r  s4 13 r 15r  s4

14  2 Use the commutative property to rewrite the expression. Use the product rule to write the expression with one radical.

Quotient rule S Product rule S 824 27  2 gives a quotient of 13 and a remainder of 1. Use the commutative property to rewrite the expression. Use the product rule to write the expression with one radical.

You Try 7 Simplify completely. Assume all variables represent nonnegative real numbers. a) 2c5d12

b) 227x10y9

c)

40u13 B v 20

6. Multiply, Divide, and Simplify Expressions Containing Square Roots At the beginning of this section we discussed multiplying and dividing radicals such as 15  12 and

200 A 2

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Section 10.3 Simplifying Expressions Containing Square Roots

651

Next we will look at some examples of multiplying and dividing radical expressions that also contain variables. Remember to always look at the result and ask yourself, “Is the radical in simplest form?” If it is not, simplify completely.

Example 8 In-Class Example 8 Perform the indicated operation and simplify completely. Assume all variables represent nonnegative real numbers. a) 18b  12b b) 23x5y  26xy4 236r7 c) 22r3 answer: a) 4b b) 3x3y2 12y c) 3r2 12

Perform the indicated operation and simplify completely. Assume all variables represent nonnegative real numbers. 220x5 a) 16t  13t b) 22a3b  28a2b5 c) 15x

Solution a) 16t  13t  16t  3t  218t2  118  2t2  19  2  t  19  12  t  3 12  t  3t12

Product rule Product rule 9 is a perfect square; 2t2  t Product rule 19  3 Use the commutative property to rewrite the expression.

b) 22a b  28a2b5 There are two good methods for multiplying these radicals. i) Multiply the radicands to obtain one radical. 3

22a3b  28a2b5  22a3b  8a2b5  216a5b6

Product rule Multiply.

Is the radical in simplest form? No.  116  2a5  2b6  4  a2 1a  b3  4a2b3 1a

Product rule Evaluate. Commutative property

ii) Simplify each radical, then multiply. 22a3b  12  2a3  1b  12  a1a  1b  a 12ab

28a2b5  18  2a2  2b5  212  a  b2 1b  2ab2 12b

Then, 22a3b  28a2b5  a 12ab  2ab2 12b  a  2ab2  12ab  12b  2a2b2 24ab2  2a2b2  14  1a  2b2  2a2b2  2  1a  b  2a2b2  2  b  1a  4a2b3 1a Both methods give the same result.

Commutative property Multiply. Product rule Evaluate. Commutative property Multiply.

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652

Chapter 10 Radicals and Rational Exponents

c)

220x5 can be simplified two different ways. 15x i) Begin by using the quotient rule to obtain one radical and simplify the radicand. 220x5 20x5  B 5x 15x  24x4  14  2x4  2x2

Quotient rule Simplify. Product rule Simplify.

ii) Begin by simplifying the radical in the numerator. 220x5 120  2x5  15x 15x 14  15  x2 1x  15x 2 15  x2 1x  15x 2x2 15x  15x  2x2

Product rule Product rule; simplify 2x5. 14  2 Product rule Divide out the common factor.

Both methods give the same result. In this case, the second method was longer. Sometimes, however, this method can be more efficient.

You Try 8 Perform the indicated operation and simplify completely. Assume all variables represent nonnegative real numbers. a) 22n3  16n

b) 215cd5  23c2d

c)

2128k9 22k

Answers to You Try Exercises 1) a) 130 b) 110r 4) a) y5

b) 12p8

7) a) c2d6 1c

c)

2) a) 2 17 315 w2

b) 3x5y4 13y

b) 513

5) a) m2 1m c)

2u6 110u v10

c) 6 12

b) z9 1z

3) a)

10 13

6) a) m6 1m

8) a) 2n2 13

b) 3

c) 5 12 d)

b) 10v3 1v

b) 3cd3 15c

c) 8k4

111 6

c) 4a 12a

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Section 10.3 Simplifying Expressions Containing Square Roots

10.3 Exercises

 Practice Problems

 NetTutor  Self-Test

 e-Professors

653

 Videos

Objective 1

video

Unless otherwise stated, assume all variables represent nonnegative real numbers. Multiply and simplify. 1) 13  17

121

2) 111  15

155

3) 110  13

130

4) 17  12

114

5) 16  1y

16y

6) 15  1p

15p

Objective 2

Label each statement as true or false. Give a reason for your answer.

60 A5

36)

40 A5

212

37)

1120 16

38)

154 13

2 15

312

39)

130 12

115

40)

135 15

17

41)

6 A 49

16 7

42)

2 A 81

12 9

43)

45 A 16

315 4

44)

60 A 49

2115 7

2 13

7) 120 is in simplest form.

Objective 5

8) 135 is in simplest form.

Simplify completely. Assume all variables represent positive real numbers.

9) 142 is in simplest form.

45) 2x8

x4

46) 2q6

q3

10) 163 is in simplest form.

47) 2m10

m5

48) 2a2

a

Objectives 2 and 3

49) 2w

50) 2t

t8

False; 20 contains a factor of 4 which is a perfect square.

False; 63 contains a factor of 9 which is a perfect square.

Simplify completely. If the radical is already simplified, then say so. video

35)

11) 120

2 15

12) 112

2 13

13) 154

3 16

14) 163

3 17

15) 133

simplified

16) 115

simplified

17) 18

2 12

18) 124

2 16

19) 180

4 15

20) 1108

6 13

21) 198

7 12

22) 196

4 16

23) 138

simplified

24) 146

simplified

25) 1400

20

26) 1900

video

31)

14 149 8 A2

154 33) 16

52) 29z8

3z4

53) 264k6

8k3

54) 225p20

5p10

55) 228r4

2r2 17

56) 227z12

3z6 13

57) 2300q22

10q11 13

58) 250n4

5n2 12

9

60)

63)

81 A c6

c3

140

2 110

2t

t

75 A y12

5 13

8

4

y6

62) 64)

h2 B 169

h 13

118

312

2m30

m15

44 A w2

2111 w

Objective 6

Simplify completely.

Simplify completely.

29)

10c

61)

Objective 4

144 27) A 25

w

16

7

51) 2100c2

59)

30

14

12 5

16 28) A 81

2 7

30)

2

3

164 1121

4 9 8 11

75 A3

5

148 34) 13

4

32)

video

65) 2a5

a2 1a

66) 2c7

c3 1c

67) 2g13

g6 1g

68) 2k15

k7 1k

69) 2b25

b12 1b

70) 2h31

h15 1h

71) 272x3

6x12x

72) 2100a5

10a2 1a

73) 213q7

q3 113q

74) 220c9

2c4 15c

75) 275t11

5t5 13t

76) 245p17

3p8 15p

77) 2c8d2

c4d

78) 2r4s12

r2s6

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654

Chapter 10 Radicals and Rational Exponents

79) 2a4b3

a2b1b

80) 2x2y9

xy4 1y

97) 1w  2w5

w3

81) 2u5v7

u2v3 1uv

82) 2f 3g9

fg4 1fg

99) 2n3  2n4

n3 1n

100) 2a10  2a3

a6 1a

83) 236m9n4

6m4n2 1m

84) 24t6u5

2t3u2 1u

101) 12k  28k5

4k3

102) 25z9  25z3

5z6

85) 244x12y5

2x6y2 111y

86) 263c7d 4

3c3d 2 17c

103) 26x4y3  22x5y2

104) 25a6b5  210ab4

87) 232t5u7

4t2u3 12tu

88) 2125k 3l 9 5kl 4 15kl

105) 28c9d2  25cd7

106) 26t3u3  23t7u4

89)

a7 B 81b6

a3 1a

91)

3r9 B s2

r4 13r s

9b3

90)

x5 B 49y6

x2 1x

92)

17h11 B k8

h5 117h

video

7y3

4

k

107) 109)

98) 2d3  2d11

218k11 22k3

108)

3k4

2120h8 23h

2

110)

2h3 110

250a

16

Objective 7

111)

Perform the indicated operation and simplify. Assume all variables represent positive real numbers. 93) 15  110 5 12

94) 18  16

95) 121  13 3 17

96) 12  114 217

d7

248m15 23m9

4m3

272c10 26c2

2c4 13

221z

18

25a

7

112)

a 110a 4

23z13

z2 17z

413

Section 10.4 Simplifying Expressions Containing Higher Roots Objectives 1. Multiply Higher Roots 2. Simplify Higher Roots of Integers 3. Use the Quotient Rule for Higher Roots 4. Simplify Radical Expressions Containing Variables 5. Multiply and Divide Radicals with Different Indices

Example 1 In-Class Example 1 Multiply. Assume the variable represent a nonnegative real number. 4 4 3 3 a) 1 3 1 8 b) 1 n 1 21 4 3 answer: a) 1 24 b) 1 21

4 In Section 10.1 we first discussed finding higher roots like 1 16  2 and 3 1 27  3. In this section, we will extend what we learned about multiplying, dividing, and simplifying square roots to doing the same with higher roots.

1. Multiply Higher Roots Definition

Product Rule for Higher Roots If a and b are real numbers such that the roots exist, then n

n

n

1a  1b  1a  b

This rule enables us to multiply and simplify radicals with any index in a way that is similar to multiplying and simplifying square roots.

Multiply. Assume the variable represents a nonnegative real number. 3 3 4 4 a) 1 2  1 7 b) 1 t  1 10

Solution 3 3 3 3 a) 1 2 1 7 1 27 1 14 4 4 4 4 b) 1 t  1 10  1 t  10  1 10t

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Section 10.4 Simplifying Expressions Containing Higher Roots

655

You Try 1 Multiply. Assume the variable represents a nonnegative real number. 4 4 a) 1 6  15

b)

5

5

1 8  2k2

Remember that we can apply the product rule in this way only if the indices of the radicals are the same. Later in this section we will discuss how to multiply radicals with different indices.

2. Simplify Higher Roots of Integers In Section 10.3 we said that one condition that must be met for a square root to be simplified is that its radicand does not contain any perfect squares. Likewise, to be considered in simplest form, the radicand of an expression containing a cube root cannot contain any perfect cubes, the radicand of an expression containing a fourth root cannot contain any perfect fourth powers, and so on. Next we list the conditions that determine when a radical with any index is in simplest form.

When Is a Radical Simplified? n

Let P be an expression and let n be a positive real number.Then 1 P is completely simplified when all of the following conditions are met: 1)

The radicand does not contain any factors (other than 1) which are perfect n th powers.

2)

The radicand does not contain any fractions.

3)

There are no radicals in the denominator of a fraction.

Condition 1) implies that the radical cannot contain variables with exponents greater than or equal to n, the index of the radical.

To simplify radicals with any index, we reverse the process of multiplying radicals where a or b is an nth power. n

n

n

1a  b  1a  1b Remember, to be certain that a radical is simplified completely always look at the radical carefully and ask yourself, “Is the radical in simplest form?”

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656

Chapter 10 Radicals and Rational Exponents

Example 2 In-Class Example 2 Simplify completely. 3 4 a) 1 81 b) 1 144 3 4 answer: a) 31 3 b) 21 9

Simplify completely. 3 4 a) 1 b) 1 250 48

Solution 3 a) We will look at two methods for simplifying 1 250. i) Since we must simplify the cube root of 250, think of two numbers that multiply to 250 so that at least one of the numbers is a perfect cube. 250  125  2 3

Rewrite 1250: 3

3

1250  1125  2 3 3  1125  12 3  5 12

125 is a perfect cube. Product rule 3 1 125  5

3 Is 5 12 in simplest form? Yes, because 2 does not have any factors that are perfect cubes. ii) Begin by using a factor tree to find the prime factorization of 250.

250  2  53 3 Rewrite 1250 using the product rule. 3 3 1 250  2 2  53 3 3 3  12  2 5 3  12  5 3  51 2

2  53 is the prime factorization of 250. Product rule 3 3 2 5 5 Commutative property

We obtain the same result using either method. 4 b) We will use two methods for simplifying 1 48. i) Since we must simplify the fourth root of 48, think of two numbers that multiply to 48 so that at least one of the numbers is a perfect fourth power. 48  16  3 4

Rewrite 148: 4 4 1 48  1 16  3 4 4 1 16  1 3 4  2 13

16 is a perfect fourth power. Product rule 4 1 16  2

4 Is 21 3 in simplest form? Yes, because 3 does not have any factors that are perfect fourth powers. ii) Begin by using a factor tree to find the prime factorization of 48.

48  24  3 4 Rewrite 148 using the product rule. 4 4 4 1 48  2 2 3 4 4 4  22  1 3 4  21 3

24  3 is the prime factorization of 48. Product rule 4 224  2

Once again, both methods give us the same result.

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Section 10.4 Simplifying Expressions Containing Higher Roots

657

You Try 2 Simplify completely. 3 a) 1 40

5 b) 1 63

3. Use the Quotient Rule for Higher Roots Definition

Quotient Rule for Higher Roots If a and b are real numbers such that the roots exist and b  0, then n

a 1a  n . Ab 1b n

We apply the quotient rule when working with nth-roots the same way we apply it when working with square roots.

Example 3 In-Class Example 3 Simplify completely. 3 48 1 56 a) 4 b) 3 A3 12 3 answer: a) 2 b) 1 28

Simplify completely. 81 3  a) b) A 3

3 1 96 3 1 2

Solution 81 81 81 81 . Let’s think of it as . as or 3 3 3 3 81 Neither 81 nor 3 is a perfect cube. But if we simplify we get 27, which 3 is a perfect cube.

a) We can think of 

81 3 1 27 A 3  3 3



Simplify 

81 . 3

b) Let’s begin by applying the quotient rule to obtain a fraction under one radical and simplify the fraction. 3

196 3

12



3 96 A2

Quotient rule 96 . 2 8 is a perfect cube.

3 1 48

Simplify

3 1 86 3 3 1 8 1 6 3  2 16

Product rule 3 1 82

3 Is 216 in simplest form? Yes, because 6 does not have any factors that are perfect cubes.

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658

Chapter 10 Radicals and Rational Exponents

You Try 3 Simplify completely. a)

1 A 81 4

b)

3 1 162 3 1 3

4. Simplify Radical Expressions Containing Variables In Section 10.2 we discussed the relationship between radical notation and fractional exponents. Recall that

If a is a nonnegative number and m and n are integers such that n is positive, then 2am  amn. n

That is, the index of the radical becomes the denominator of the fractional exponent, and the power in the radicand becomes the numerator of the fractional exponent.

This is the principal we use to simplify radicals with indices greater than 2.

Example 4 In-Class Example 4 Simplify completely. Assume all variables represent nonnegative real numbers. 4 12 3 a) 2 b) 2 x 8m9n6 12 f c) 6 24 Bg answer: a) x3 b) 2m3n2 f2 c) 4 g

Simplify completely. Assume all variables represent nonnegative real numbers. c10 3 15 4 a) 2 b) 2 c) 5 30 y 16t24u8 Bd

Solution 3 15 a) 2 y  y15/3  y5 4 4 4 24 4 8 b) 2 16t24u8  1 16  2 t 2 u 244 84 2t u  2t6u2 5 10 c10 2 c c) 5 30  5 30 Bd 2d c10/5  30/5 d c2  6 d

Product rule Write with a rational exponent. Divide. Quotient rule Write with a rational exponent. Divide.

You Try 4 Simplify completely. Assume all variables represent nonnegative real numbers. 3 3 21 a) 2 ab

b)

m12 B 16n20 4

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Section 10.4 Simplifying Expressions Containing Higher Roots

659

To simplify a radical expression if the power in the radicand does not divide evenly by the index, we use the same methods we used in Section 10.3 for simplifying similar expressions with square roots. We can use the product rule or we can use the idea of quotient and remainder in a division problem.

Example 5 In-Class Example 5 4 31 Simplify 2 d completely in two ways. 4 3 answer: d7 2 d

4

Simplify 2x23 completely in two ways: 1) use the product rule and 2) divide the exponent by the index and use the quotient and remainder. Assume the variable represents a nonnegative real number.

Solution 1) Using the product rule: 23 4 To simplify 2x23, write x as the product of two factors so that the exponent of one of the factors is the largest number less than 23 that is divisible by 4 (the index). 4 23 4 20 2 x 2 x  x3 4 20 4 2 x 2 x 4  x20/4  1x 4  x5 1 x

20 is the largest number less than 23 that is divisible by 4. Product rule Use a fractional exponent to simplify. 20  4  5

2) Using the quotient and remainder: 5 d Quotient 4 23 To simplify 2 x , divide 4 23 20 3 d Remainder Recall from our work with square roots in Section 10.3 that 1) the exponent on the variable outside of the radical will be the quotient of the division problem. and 2) the exponent on the variable inside of the radical will be the remainder of the division problem. 4 23 4 3 2 x  x5 2 x 4

Is x5 2x3 in simplest form? Yes, because the exponent inside of the radical is less than the index.

You Try 5 5

Simplify 2r32 completely using both methods shown in Example 5. Assume r represents a nonnegative real number.

We can apply the product and quotient rules together with the methods above to simplify certain radical expressions.

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Chapter 10 Radicals and Rational Exponents

Example 6 In-Class Example 6 4 Simplify 2 32p5q8 completely. Assume the variables represent positive real number. 4 answer: 2pq2 1 2p

3 Completely simplify 2 56a16b8. Assume the variables represent nonnegative real numbers.

Solution 3 3 3 16 3 8 2 56a16b8  1 56  2 a 2 b 3 3 3 2 5 3 1  18  17  a 2a  b2 2 b

c

c Product 16  3 gives a rule quotient of 5 and a remainder of 1.

Product rule

c

660

8  3 gives a quotient of 2 and a remainder of 2.

3 3 3 2  21 7  a5 1 a  b2 2 b 3 3 3 2 5 2  2a b  17  1a  2 b

3  2a5b2 2 7ab2

3 Simplify 1 8.

Use the commutative property to rewrite the expression. Product rule

You Try 6 Simplify completely. Assume the variables represent nonnegative real numbers. 4 a) 2 48x15y22

b)

r19 B 27s12 3

5. Multiply and Divide Radicals with Different Indices The product and quotient rules for radicals apply only when the radicals have the same indices. To multiply or divide radicals with different indices, we first change the radical expressions to rational exponent form.

Example 7 In-Class Example 7 Perform the indicated operation, and write the answer in simplest radical form. Assume the variables represent positive real numbers. 4 2b3 4 a) 2m5  1m b) 3 1b 4 answer: a) m2m3 12 b) 2 b5

Multiply the expressions, and write the answer in simplest radical form. Assume the variables represent positive real numbers. 3 2 2 x  1x

Solution 3 The indices of 2x2 and 1x are not the same, so we cannot use the product rule right now. Rewrite each radical as a fractional exponent, use the product rule for exponents, then convert the answer back to radical form. 3 2 2 x  1x  x23  x12  x46  x36 4 3  x6  6 7/6 x 6 7 2 x 6  x1 x

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Section 10.4 Simplifying Expressions Containing Higher Roots

661

You Try 7 Perform the indicated operation, and write the answer in simplest radical form. Assume the variables represent positive real numbers. 4 6 a) 1 y  1y

3 2 2 c 1c

b)

Answers to You Try Exercises 4 1) a) 1 30

5 5) r6 2r2

10.4 Exercises

5 b) 2 8k2

3 2) a) 21 5

4 6) a) 2x3y5 23x3y2

b)

b) simplified 3 r6 1 r

3) a)

3 b) 3 1 2

4) a) ab7 b)

m3 2n5

12

6 7) a) 2 y5 b) 1c

4

3s

 Practice Problems

1 3

 NetTutor  Self-Test

 e-Professors

 Videos

23)

25)

3) How do you know that a radical expression containing a cube root is completely simplified?

27)

4) How do you know that a radical expression containing a fourth root is completely simplified?

5 5 7) 29  2m2

5 2 9m2

3 3 9) 2a2  2b

3 2a2b

5 5 6) 1 6  1 2

5 1 12

4 4 8) 211  2h3 5 5 10) 2t2  2u4

Objectives 2 and 3

Simplify completely. video

3 11) 1 24

3 21 3

3 12) 1 48

3 21 6

4 13) 1 64

4 21 4

4 14) 1 32

4 21 2

3 15) 1 54

3 31 2

3 16) 1 88

3 21 11

3 17) 1 2000

3 10 1 2

3 18) 1 108

3 31 4

5 19) 1 64

5 21 2

4 20) 1 162

4 31 2

1 21) B 16

1 2

1 22) B 125

1 5

4

3

4 2 11h3 5

2t2u4

54 2

3 2 48 3 2 2 4 2 240 4 2 3

3

24)

3 21 3

26)

4

28)

215

4 48 B3 3 2 500 3 2 2 3 2 8000 3 2 4

2

3 51 2

3

10 1 2

Simplify completely. Assume all variables represent positive real numbers.

Multiply. Assume the variable represents a nonnegative real number. 3 1 20



Objective 4

Objective 1

3 3 5) 1 5  1 4

3

B

video

3 29) 2d 6

d2

3 30) 2g9

g3

4 31) 2n20

n5

4 32) 2t36

t9

5 33) 2x5y15

xy3

6 34) 2a12b6

a2b

3 35) 2w14

3 2 w4 2 w

3 36) 2b19

3 b6 2 b

4 37) 2y9

4 y2 1 y

4 38) 2m7

4 m2 m3

3 39) 2d5

3 2 d2 d

3 40) 2c29

3 2 c9 2 c

3 41) 2u10v15

3 u3v5 1 u

3 42) 2x9y16

3 x3y5 1 y

3 43) 2b16c5

3 b5c2 bc2

4 44) 2r15s9

4 3 r3s2 2 rs

4 45) 2m3n18

4 n4 2 m3n2

3 46) 2a11b

3 2 a3 2 ab

3 47) 224x10y12

3 2x3y4 1 3x

3 48) 254y10z24

3 3y3z8 2 2y

3 3 49) 2250w4x16 5wx5 13 2wx 50) 272t17u7

51)

m8 B 81 4

m2 3

52)

16 B x12 4

3 2t5u2 2 9t2u

2 x3

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662

video

Chapter 10 Radicals and Rational Exponents

53)

32a23 B b15

5 3 2a4 2 a

55)

t9 B 81s24

4 t2 2 t

57)

u28 B v3

3 u9 1 u v

5

b

4

3

3s6

3

54)

h17 B 125k 21

3 2 h5 2 h

56)

32c9 B d20

5 4 2c2 c

58)

m13 B n8

4 m3 2 m

3

7

5k

5

d4

4

2

n

3 21 3

3 3 60) 1 4  1 10

3 21 5

3 3 61) 1 9  1 12

3 31 4

3 3 62) 1 9  1 6

3 31 2

3 3 63) 1 20  1 4

3 21 10

3 3 64) 1 28  1 2

3 21 7

3 3 65) 2m4  2m5

m3

3 3 66) 2t5  2t

4 4 67) 2k7  2k9

k4

3 3 69) 2r7  2r4

3 2 r3 2 r

5

71) 2p video

3

14

73) 29z

5

 2p

11

9

3

4 5

p 2p

 23z3

8

3z6 1 z

3

h14 B h2

h4

76)

a20 B a14

a2

77)

c11 B c4

3 c2 1 c

78)

z16 B z5

3 2 z3 2 z

79)

162d21 4 3 3d4 2 d B 2d2

80)

48t11 B 3t6

4 2t1 t

3

3

4

3

3

4

Objective 5

Perform the indicated operation and simplify. Assume the variables represent positive real numbers. 3 3 59) 1 6  1 4

75)

The following radical expressions do not have the same indices. Perform the indicated operation, and write the answer in simplest radical form. Assume the variables represent positive real numbers. video

3 81) 1p  1 p

6 5 2 p

3 4 82) 2y2  2y

2 y11

t2

4 83) 2n3  2n

4 n1 n

5 84) 2k4  2k

k2 k3

4 4 68) 2a9  2a11

a5

5 3 85) 2c3  2c2

c2 c4

3 5 86) 2a2  2a2

a1 a

3 3 70) 2y2  2y17

3 y6 1 y

4 1 w

88)

15

90)

5

17

3

4

72) 2c

5

 2c 3

9

2w

87)

4

2w

5 5

c 1c

74) 22h  24h

16

3 2 2h6 2 h

5 4 2 t

89)

15

2 t2

3 2 2 t

4 2 m3

2m 4 3 2 h 3 2 2 h

12

10

15

4 1 m

15

2h

Just as we can add and subtract like terms such as 4x  6x  10x we can add and subtract like radicals such as 4 13  6 13 Like radicals have the same index and the same radicand.

Some examples of like radicals are 4 23 and 623,

3 3 2 5 and 82 5,

2x and 72x,

3 2 3 2 22 a b and 2 ab

We add and subtract like radicals in the same way we add and subtract like terms—add or subtract the “coefficients” of the radicals and multiply that result by the radical. Recall that we are using the distributive property when we are combining like terms in this way.

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Example 1 In-Class Example 1 Add. a) 3x  11x b) 9 15  215 answer: a) 14x b) 1115

b)

663

4 13  6 13

Solution a) First notice that 4x and 6x are like terms. Therefore, they can be added. 4x  6x  (4  6)x  10x

Distributive property Simplify.

Or, we can say that by just adding the coefficients, 4x  6x  10x. b) Before attempting to add 4 13 and 613, we must be certain that they are like radicals. Since they are like, they can be added. 4 13  6 23  (4  6) 13  10 13

Distributive property Simplify.

Or, we can say that by just adding the coefficients of 13, we get 4 13  6 13  10 13.

You Try 1 Add. a) 9c  7c

Example 2 In-Class Example 2 Perform the operations and simplify. Assume all variables represent nonnegative real numbers. a) 12  4  1612  19 3 3 b) 9 1y  21 y  3 1g  21 g answer: a) 15  1712 3 b) 61y  41 y

b) 9110  7110

Perform the operations and simplify. Assume all variables represent nonnegative real numbers. 3 3 a) 8  15  12  3 15 b) 6 1x  11 1 x  21x  61 x

Solution a) Begin by writing like terms together. 8  15  12  3 15  8  12  15  3 15  4  (1  3) 15  4  4 15

Is 4  4 15 in simplest form? Yes. The terms are not like so they cannot be combined further, and 15 is in simplest form. 3 b) Begin by noticing that there are two different types of radicals: 1x and 1 x. Write the like radicals together. 3 3 3 3 6 1x  111 x  2 1x  6 1 x  6 1x  2 1x  111 x  61 x Commutative 3

 (6  2) 1x  (11  6) 1 x 3  8 1x  5 1 x 3

property Distributive property

Is 81x  5 1 x in simplest form? Yes. The radicals are not like (they have different indices) so they cannot be combined further. Also, each radical, 3 1x and 1 x, is in simplest form.

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664

Chapter 10 Radicals and Rational Exponents

You Try 2 Perform the operations and simplify.Assume all variables represent nonnegative real numbers. 3 3 a) 9  1 4  2  101 4

3 3 b) 81 2n  3 12n  5 12n  5 1 2n

2. Simplify Radical Expressions Containing Integers, Then Add or Subtract Them Sometimes it looks like two radicals cannot be added or subtracted. But if the radicals can be simplified and they turn out to be like radicals, then we can add or subtract them.

Example 3 In-Class Example 3 Perform the operations and simplify. a) 127  13 b) 3 120  2112  615 4 4 c) 1 6  41 96 answer: a) 213 b) 1215  413 4 c) 51 4

1)

Write each radical expression in simplest form.

2)

Perform the operations and simplify. a) 18  5 12 b) 6 118  3 150  145

3 3 c) 7 1 40  1 5

Solution a) 18 and 5 12 are not like radicals. Can either radical be simplified? Yes. We can simplify 18. 18  5 12  14  2  5 12  14  12  5 12  2 12  5 12  7 12

4 is a perfect square. Product rule 14  2 Add like radicals.

b) 6 118, 3 150, and 145 are not like radicals. In this case, each radical should be simplified to determine if they can be combined. 6 118  3 150  145  6 19  2  3 125  2  19  5  6  19  12  3 125  12  19  15 Product rule Simplify radicals.  6  3  12  3  5  12  3 15 Multiply.  1812  15 12  3 15 Add like radicals.  3312  3 15 33 12  3 15 is in simplest form since they aren’t like expressions. 3 3 3 3 8 is a perfect cube. c) 7 1 40  1 5  7 1 8  5  1 5 3 3 3 Product rule  7 1 8  1 5  1 5 3 3 3  7  2  1 5 1 5 1 82 3 3 Multiply.  14 1 5  1 5 3 Add like radicals.  13 1 5

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You Try 3 Perform the operations and simplify. a) 713  112

b) 2163  11128  2121

3 3 c) 1 54  51 16

Example 4 In-Class Example 4 Perform the operations and simplify. Assume all variables represent nonnegative real number a) 4150n  6 12n 3 3 b) 2x3y  2y answer: a) 1412n 3 b) (x  1) 1 y

Perform the operations and simplify. Assume all variables represent nonnegative real numbers. 3 3 a) 10 18t  9 12t b) 2xy6  2x7

Solution a) 12t is simplified, but 18t is not. We must simplify 18t. 18t  18  1t  14  12  1t  2 12  1t  2 12t

Product rule 4 is a perfect square. 14  2 Product rule

Substitute 2 12t for 18t in the original expression. 10 18t  9 12t  10(2 12t)  912t  20 12t  9 12t  11 12t

Substitute 212t for 18t. Multiply. Subtract.

3 3 b) Each radical in the expression 2xy6  2x7 must be simplified. 3 3 2 xy6  y2 2 x

632

3 7 3 1 2 x  x2 2 x

7  3 gives a quotient of 2 and a remainder of 1.

Then, substitute the simplified radicals in the original expression. 3 3 7 3 3 2 xy6  2 x  y2 1 x  x2 1 x 3 2 2  1x( y  x )

Substitute 3 Factor out 1 x from each term.

3 3 In this problem we cannot add y2 1 x  x2 1 x like we added radicals in previous 3 examples, but we can factor out 1x. 3 1x( y2  x2 ) is the completely simplified form of the sum.

You Try 4 Perform the operations and simplify. Assume all variables represent nonnegative real numbers. a) 216k  4 154k

4 4 b) 2mn11  281mn3

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Chapter 10 Radicals and Rational Exponents

Answers to You Try Exercises 3 2) a) 7  11 1 4

1) a) 16c b) 16 110

3 b) 13 1 2n  212n

3

4 4) a) 1416k b) 2 mn3 (n2  3)

3) a) 5 13 b) 16 17  2 121 c) 1312

 Practice Problems

10.5 Exercises

 NetTutor  Self-Test

 e-Professors

 Videos

Objectives 1–3

39) 4d2 1d  242d5

40) 16k4 1k  132k9

They have the same index and the same radicand.

Perform the operation and simplify. Assume all variables represent nonnegative real numbers. 14 12

4) 11 17  7 17

18 17

5) 413  913

5 13

6) 16 12  2012

4 12

3

7) 714  814 video

3

3

8) 10 1 5  2 15

3

15 14

42) 8w2w5  42w7 12w3 1w

3 3 3 t 43) 9t3 1t  5 2t10 4t3 1

3 3 3 44) 8r4 1r  162r13 8r4 1r

47) 1115z  2 120z 15 15z 48) 8 13r  5 112r 213r 49) 218p  612p 212p 50) 4 163t  6 17t 1817t video

3 81 5

3 25a2 3a2

11  3113

3 3 52) 3140x  1215x

3 61 5x

10) 8  3 16  416  9

1  16

3 3 53) 4c2 2108c  15 232c7

3 18c2 1 4c

3 3 11) 152z2  20 2z2

3 2 5 2 z

3 3 54) 162128h2  4 216h2

3 722 2h2

55) 2xy3  3y2xy

4y1xy

9) 6  113  5  2 113

3

12) 71p  41p 3

5

3

3 1p 3

5

13) 22n  92n  112n  2n 2

4

2

3

3

2

4

14) 51s  31s  2 1s  41s

2

3

5

92n  10 2n

4

2

2

3

2 15c  216c video

7a1ab

57) 6c 28d  9d 22c d

3c2d 12d

58) 11v25u3  2u245uv2

5uv15u

59) 3275m n  m112mn

17m13mn

2

16) 1012m  6 13m  12m  813m 9 12m  14 13m 17) 613  112

56) 5a2ab  22a b 3

91s  1 s

15) 15c  8 16c  15c  6 16c

3

4

3

413

18) 145  415

7 15

19) 148  13

513

20) 144  8111

6 111

61) 18a 27a b  2a 27a b

3 20a5 2 7a2b

21) 120  4145

1415 22) 128  2163

4 17

3 3 62) 8p2q211pq2  3p2 288pq5

3 14p2q 2 11pq2

23) 3198  4 150 4112 24) 3150  4 18

7 12

4 4 63) 15cd 29cd  29c5d5

14cd 19cd

25) 132  318

10 16

4 4 64) 7yz2 211y4z  3z211y8z5

4 10y2z2 1 11z

212 26) 3124  196

60) y254xy  6 224xy

3

5 3

2

3 3

9y16xy 8

4

27) 112  175  13

613

65) 2m5  2mn2

1m(m2  n)

28) 12  198  150

12

66) 2z3  2y6z

1z(z  y3 )

3 3 67) 2a9b  2b7

3 1b(a3  b2 )

3 3 68) 2c8  2c2d3

3 2 2 2 c (c  d)

3 3 69) 2u2v6  2u2

2u2 (v2  1)

29) 120  2145  180

815

30) 196  124  5154

916

3

3

3

31) 819  172 3

3

3 101 9 32) 5188  2 111

3

3

3

33) 2181  14 13 8 13 34) 613  3181 3

3

3

35) 16  148 video

41) 92n5  4n2n3 5n2 1n

3 3 51) 7281a5  4a23a2

3

video

3k4 1k

4 4 4 11 4 2 4 3 4 3 a 46) 3 2c  6c 2c 3c2 2c3 45) 5a2a7  2a11 6a2 2

3) 512  912 3

20d2 1d

37) 6q1q  7 2q

3

16 3

3

3

36) 11 116  712

3 12 1 11 3

3 13 3

29 12

13q1q 38) 11 2m  8m 1m 19m1m 3

3

3

70) 1s  2r6s4

3

3 1s(1  r2s)

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Section 10.6 Combining Multiplication, Addition, and Subtraction of Radicals Objectives 1. Multiply a Binomial Containing Radical Expressions by a Monomial 2. Multiply Two Binomials Containing Radical Expressions Using FOIL 3. Square a Binomial Containing Radical Expressions 4.

Multiply Two Binomials of the Form (a  b)(a  b) Containing Radicals

In-Class Example 1 Multiply and simplify. Assume all variables represent nonnegative real numbers. a) 3( 17  163) b) 13( 150  120) c) 1m(3 1m  118n) answer: a) 1217 b) 516  2115 c) 3m  312mn

In Section 10.3 we learned to multiply radicals like 16  12. In this section, we will learn how to simplify expressions that combine multiplication, addition, and subtraction of radicals.

1. Multiply a Binomial Containing Radical Expressions by a Monomial Example 1 Multiply and simplify. Assume all variables represent nonnegative real numbers. a) 4( 15  120) b) 12( 110  115) c) 1x( 1x  132y)

Solution a) Since 120 can be simplified, we will do that first. 120  14  5  14  15  2 15

4 is a perfect square. Product rule 14  2

Substitute 2 15 for 120 in the original expression. 4( 15  120)  4( 15  2 15)  4(15)  4 15

Substitute 2 15 for 120. Subtract. Multiply.

b) Neither 110 nor 115 can be simplified. Begin by applying the distributive property. 12( 110  115)  12  110  12  115  120  130

Distribute. Product rule

Is 120  130 in simplest form? No. 120 can be simplified.  14  5  130  14  15  130  2 15  130

4 is a perfect square. Product rule 14  2

2 15  130 is in simplest form. They are not like radicals, so they cannot be combined. c) Since 132y can be simplified, we will do that first. 132y  132  1y  116  2  1y  116  12  1y  4 12y

Product rule 16 is a perfect square. Product rule 116  4; multiply 12  1y.

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Chapter 10 Radicals and Rational Exponents

Substitute 4 12y for 132y in the original expression. 1x( 1x  132y)  1x( 1x  412y)  1x  1x  1x  4 12y  x  412xy

Substitute 412y for 132y. Distribute. Multiply.

You Try 1 Multiply and simplify. Assume all variables represent nonnegative real numbers. a) 6( 175  2 13)

b) 13( 13  121)

c) 1c( 2c3  2100d)

2. Multiply Two Binomials Containing Radical Expressions Using FOIL In Chapter 6, we first multiplied binomials using FOIL (First Outer Inner Last). (2x  3)(x  4)  2x  x  2x  4  3  x  3  4 F O I L 2  2x  8x  3x  12  2x2  11x  12 We can multiply binomials containing radicals the same way.

Example 2 In-Class Example 2 Multiply and simplify. Assume all variables represent nonnegative real numbers. a) (3  13)(6  13) b) (4 12  15) (212  215) c) ( 15  12t) ( 15  3 12) answer: a) 21  913 b) 6  6110 c) S  415t  6t

Multiply and simplify. Assume all variables represent nonnegative real numbers. a) (2  15)(4  15) b) (2 13  12)( 13  512) c) ( 1r  13s)( 1r  8 13s)

Solution a) Since we must multiply two binomials, we will use FOIL. (2  15)(4  15)  2  4  2  15  4  15  15  15 F O I L 5 Multiply.  8  2 15  415   13  6 15 Combine like terms. b) (213  12)( 13  5 12)  2 13  ( 13)  2 13  (5 12)  12  ( 13)  12  (512) F O I L  23  (10 16)  16  (5  2) Multiply. Multiply.  6  10 16  16  10 Combine  4  916 like terms.

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c) ( 1r  13s)( 1r  8 13s)  1r  ( 1r)  1r  (8 13s)  13s  ( 1r)  13s  (813s) F O I L  r  813rs  13rs  8  3s  r  813rs  1 13rs  24s  r  913rs  24s

Multiply. Multiply. Combine like terms.

You Try 2 Multiply and simplify. Assume all variables represent nonnegative real numbers. a) (6  17)(5  17)

b) ( 12  415) (3 12  15)

c) ( 16p  12q)( 16p  312q)

3. Square a Binomial Containing Radical Expressions Recall, again, from Chapter 6, that we can use FOIL to square a binomial or we can use these special formulas: (a  b) 2  a2  2ab  b2 (a  b) 2  a2  2ab  b2 For example, (k  7) 2  (k) 2  2(k)(7)  (7) 2  k2  14k  49 and (2p  5) 2  (2p) 2  2(2p)(5)  (5) 2  4p2  20p  25 To square a binomial containing radicals, we can either use FOIL or we can use the formulas above. Understanding how to use the formulas to square a binomial will make it easier to solve radical equations in Section 10.7.

Example 3 In-Class Example 3 Multiply and simplify. Assume all variables represent nonnegative real numbers. a) ( 15  2) 2 b) (3 12  4) 2 c) ( 12t  16) 2 answer: a) 9  415 b) 34  2412 c) 2t  413t  6

Multiply and simplify. Assume all variables represent nonnegative real numbers. a) ( 110  3) 2 b) (2 15  6) 2 c) ( 1x  17) 2

Solution a) Use (a  b) 2  a2  2ab  b2. ( 110  3) 2  ( 110) 2  2( 110)(3)  (3) 2  10  6 110  9  19  6 110

Substitute 110 for a and 3 for b. Multiply. Combine like terms.

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Chapter 10 Radicals and Rational Exponents

b) Use (a  b) 2  a2  2ab  b2. (2 15  6) 2  (215) 2  2(2 15)(6)  (6) 2  (4  5)  (4 15)(6)  36  20  24 15  36  56  24 15

Substitute 215 for a and 6 for b. Multiply. Multiply. Combine like terms.

c) Use (a  b) 2  a2  2ab  b2.

Substitute 1x for a and 17 for b. Square; product rule

( 1x  17) 2  ( 1x) 2  2( 1x)( 17)  ( 17) 2  x  2 17x  7

You Try 3 Multiply and simplify. Assume all variables represent nonnegative real numbers. a) ( 16  5) 2

b) (3 12  4) 2

c) ( 1w  111) 2

4. Multiply Two Binomials of the Form (a  b) (a  b) Containing Radicals We will review one last rule from Chapter 6 on multiplying binomials. We will use this in Section 10.7 when we divide radicals. (a  b)(a  b)  a2  b2 For example, (t  8)(t  8)  (t) 2  (8) 2  t2  64 The same rule applies when we multiply binomials containing radicals.

Example 4 In-Class Example 4 Multiply and simplify. Assume all variables represent nonnegative real numbers. a) (3  17)(3  17) b) ( 15  22)( 15  12) c) ( 1r  15)( 1r  15) answer: a) 2 b) 3 c) r  5

Multiply and simplify. Assume all variables represent nonnegative real numbers. a) (2  15)(2  15) b) ( 1x  1y)( 1x  1y)

Solution a) Use (a  b)(a  b)  a2  b2. (2  15)(2  15)  (2) 2  ( 15) 2 45  1

Substitute 2 for a and 15 for b. Square each term. Subtract.

b) Use (a  b)(a  b)  a2  b2. ( 1x  1y)( 1x  1y)  ( 1x) 2  ( 1y) 2 xy

Substitute 1x for a and 1y for b. Square each term.

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In each case, when we multiply expressions containing square roots of the form (a  b)(a  b) the radicals are eliminated. This will always be true.

You Try 4 Multiply and simplify. Assume all variables represent nonnegative real numbers. a) (4  110) (4  110)

b) ( 15h  1k) ( 15h  1k)

Answers to You Try Exercises 1) 2) 3) 4)

10.6 Exercises

a) a) a) a)

42 13 b) 3  317 c) c2  101cd 23  17 b) 26  13 110 c) 6p  813pq  6q 31  1016 b) 34  24 12 c) w  2 111w  11 6 b) 5h  k

 Practice Problems

 NetTutor  Self-Test

 e-Professors

 Videos

Objective 1

23) What formula can be used to multiply (a  b) (a  b)  a2  b2 (5  16)(5  16)?

Multiply and simplify. Assume all variables represent nonnegative real numbers. 1) 3(x  5)

2) 8(k  3)

3) 7( 16  2)

4) 5(4  17)

5) 110( 13  1)

6) 12(9  111)

7) 6( 132  12)

8) 10( 112  13)

9) 4( 145  120)

10) 3( 118  150)

11) 15( 124  154)

12) 12( 120  145)

13) 13(4  16)

14) 15( 115  12)

15) 17( 124  15)

16) 118( 18  8)

17) 1t( 1t  181u)

18) 1s( 112r  17s)

19) 1ab( 15a  127b)

20) 12xy( 12y  y1x)

3x  15

7 16  14

video

130  110

3012 4 15

130

4 13  3 12

2 142  135 t  9 1tu

a 15b  3b 13a

24) What happens to the radical terms whenever we multiply (a  b)(a  b) where the binomials contain square roots? The radicals are eliminated.

8k  24

20  517

Multiply and simplify. Assume all variables represent nonnegative real numbers.

9 12  122

25) ( p  7)( p 2 6)

26) (z  8)(z  2) 2

27) (6  17)(2  17)

28) (3  15)(1  15)

5110

29) ( 12  8)( 12  3)

30) ( 16  7)( 16  2)

513  110

31) ( 15  413)(2 15  13)

22  9115

12  24 12

32) (512  13)(2 13  12)

16  1116

213rs  s17

33) (316  212)( 12  516)

86  1413

2y1x  xy 12y

34) (2110  312)( 110  212)

10 13

2412

video

19  8 17

22  512

Objectives 2– 4

21) How are the problems Multiply (x  8)(x  3) and Multiply (3  12)(1  12) similar? What method can be used to multiply each of them? 22) How are the problems Multiply (y  5)2 and Multiply ( 17  2)2 similar? What method can be used to multiply each of them?

p  13p  42

video

z  6z  16

8  415 8  5 16

8  2 15

35) (5  2 13)( 17  12)

517  512  2 121  2 16

36) ( 15  4)( 13  6 12)

115  6110  4 13  24 12

37) ( 1x  12y)( 1x  5 12y)

x  612xy  10y

38) ( 15a  1b)(3 15a  1b)

15a  415ab  b

39) ( 16p  21q)(8 1q  516p) 40) (4 13r  1s)(3 1s  2 13r)

2 16pq  30p  16q 1013rs  3s  24r

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Chapter 10 Radicals and Rational Exponents

41) (5y2  4) 2

42) (3k  2) 2 9k2  12k  4

65) ( 1c  1d)( 1c  1d)

cd

43) ( 13  1) 2

44) (2  15) 2

66) ( 12y  1z)( 12y  1z)

2y  z

67) (5  1t)(5  1t)

25  t

68) (6  1q)(6  1q)

36  q

69) (8 1f  1g)(8 1f  1g)

64f  g

70) ( 1a  314b)( 1a  314b)

a  36b

25y  40y  16 4  213

45) ( 111  15) 16  2 155

47) (2 13  110) 22  4 130

49) ( 12  416)

9  4 15

2

46) ( 13  113)

2

48) (3 12  13)

2

16  2 139

2

21  6 16

50) ( 15  3 110)

2

95  30 12

98  16 13

51) ( 1h  17) 2

52) ( 1m  13) 2

53) ( 1x  1y)

54) ( 1b  1a)

m  2 13m  3

h  217h  7 x  21xy  y

2

2

2

Extension

a  2 1ab  b

55) (c  9)(c  9) c  81 56) (g  7)(g  7) g2  49 2

58) ( 13  1)( 13  1)

2

3 3 71) ( 1 2  3)( 1 2  3)

3 1 49

59) (6  15)(6  15)

60) (4  17)(4  17)

9

3 3 72) (1  1 6)(1  1 6)

1  1 36

7

video

Multiply and simplify.

57) ( 12  3)( 12  3) 31

3

61) (413  12)(4 13  12)

46

3 3 3 73) (1  2 1 5)(1  21 5  41 25)

62) (212  2 17)(2 12  217)

20

3 3 3 74) (3  1 2)(9  31 2  1 4)

63) ( 111  5 13)( 111  513)

64

75) [( 13  16)  12][( 13  16)  12]

7  612

64) ( 115  5 12)( 115  512)

35

76) [( 15  12)  12][( 15  12)  12]

5  2110

41

29

Section 10.7 Dividing Radicals Objectives 1. Rationalize a Denominator Containing One Square Root Term 2. Rationalize a Denominator Containing One Term That Is a Higher Root 3. Rationalize a Denominator Containing Two Terms 4. Simplify a Radical Expression by Dividing Out Common Factors From the Numerator and Denominator

It is generally agreed in mathematics that a radical expression is not in simplest form if its denominator contains a radical. 1 is not simplified 13

13 is simplified 3

This comes from the days before calculators when it was much easier to perform calculations if there were no radicals in the denominator of an expression. 13 1  . The process of eliminating radicals from the Later we will show that 3 13 denominator of an expression is called rationalizing the denominator. We will look at two types of rationalizing problems. 1) Rationalizing a denominator containing one term. 2) Rationalizing a denominator containing two terms. The way we rationalize a denominator is based on the fact that if we multiply the numerator and denominator of a fraction by the same quantity we are actually multiplying the fraction by 1. Therefore, the fractions may look different, but they are equivalent. 2 4 8   3 4 12

8 2 and are equivalent 3 12

We use the same idea to rationalize the denominator of a radical expression.

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1. Rationalize a Denominator Containing One Square Root Term The goal of rationalizing is to eliminate the radical from the denominator. With regard to square roots, recall that 1a  1a  2a2  a for a  0. For example, 12  12  222  2, 119  119  21192 2  19, 1t  1t  2t2  t 1t  02 We will use this property to rationalize the denominators of the following expressions.

Example 1 In-Class Example 1 Rationalize the denominator of each expression. 7111 2 4 a) b) c) 15 120 16 215 215 answer: a) b) 5 5 7166 c) 6

Rationalize the denominator of each expression. 5 13 1 36 a) b) c) 13 118 12

Solution 1 , ask yourself, “By 13 what do I multiply 13 to get a perfect square under the square root?” The answer is 13 since 13  13  232  19  3. Multiply by 13 in the 1 numerator and denominator. (We are actually multiplying by 1.) 13

a) To eliminate the square root from the denominator of

1 1 13   13 13 13 13  232 13  19 13  3

232  19  3

13 is in simplest form. We cannot reduce terms inside and outside of the radical. 3

Wrong:

13 131   11  1 3 31

b) Notice that we can simplify the denominator of

36 . We will do that first. 118

118  19  12  312 36 36  118 3 12 12  12

Substitute 312 for 118. Simplify

36 . 3

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Chapter 10 Radicals and Rational Exponents

12 , ask yourself, 12 “By what do I multiply 12 to get a perfect square under the square root?” The answer is 12. 12 12 Rationalize the denominator.   12 12 12 12  222 12 12  222  2 2 12  6 12 Simplify . To eliminate the square root from the denominator of

2

c) To rationalize

5 13 , multiply the numerator and denominator by 12. 12 5 13 5 13 12   12 12 12 5 16  2

You Try 1 Rationalize the denominator of each expression. a)

1 17

b)

15 127

c)

916 15

Sometimes we will apply the quotient or product rule before rationalizing.

Example 2 In-Class Example 2 Simplify completely. 6 3 22  a) b) A 20 A 11 A 9 130 16 answer: a) b) 10 3

Simplify completely. 5 3 7  a) b) A 14 A 3 A 24

Solution a) Begin by simplifying the fraction

3 . 24

3 1  A 24 A 8 11  18     

1 14  12 1 2 12 1 12  2 12 12 12 22 12 4

Simplify; quotient rule Product rule 14  2 Rationalize the denominator. 12  12  2 Multiply.

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b) Begin by using the product rule to multiply the radicands. 5 7 5 7    A 14 A 3 A 14 3

Product rule

Multiply the fractions under the radical. 1

5 7   A 14 3 2

5  A6 15  16 15 16   16 16 130  6

Multiply. Quotient rule Rationalize the denominator. Multiply.

You Try 2 Simplify completely. a)

10 A 35

b)

21 2  A 10 A 7

We work with radical expressions containing variables the same way.

Example 3 In-Class Example 3 Simplify completely. Assume all variables represent positive real numbers. 27p4 10xy 7 a) b) c) B q3 A x4y 2 1m 3p2 1q 71m answer: a) b) m q2 110xy c) xy

Simplify completely. Assume all variables represent positive real numbers. 2 12m3 6cd2 a) b) c) B 7n B cd3 1x

Solution a) Ask yourself, “By what do I multiply 1x to get a perfect square under the square root?” The perfect square we want to get is 2x2. 1x  1?  2x2  x 1x  1x  2x2  x 2 2 1x   1x 1x 1x 2 1x  2x2 21x  x

Rationalize the denominator. Multiply. 2x2  x

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Chapter 10 Radicals and Rational Exponents

b) Before rationalizing, apply the quotient rule, and simplify the numerator. 12m3 212m3  B 7n 17n 2m13m  17n

Quotient rule Simplify 212m3.

Rationalize the denominator. “By what do I multiply 17n to get a perfect square under the square root?” The perfect square we want to get is 272n2 or 249n2. 17n  1?  272n2  7n 17n  17n  272n2  7n 12m3 2m13m  B 7n 17n 2m13m 17n   17n 17n 2m121mn  7n c) Before rationalizing

Rationalize the denominator. Multiply.

6cd2 , simplify the radicand. B cd3

6cd2 6 3  B cd Ad 16  1d 16 1d   1d 1d 16d  d

Quotient rule for exponents. Quotient rule for radicals. Rationalize the denominator. Multiply.

You Try 3 Simplify completely. Assume all variables represent positive real numbers. a)

5 1p

b)

18k5 B 10m

c)

20r3s B s2

2. Rationalize a Denominator Containing One Term That Is a Higher Root Many students assume that to rationalize denominators like we have up until this point, all you have to do is multiply the numerator and denominator of the expression by the denominator as in 4 13 4 13 4    3 13 13 13

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We will see, however, why this reasoning is incorrect. 4 To rationalize an expression like we asked ourselves, “By what do I multiply 13 13 to get a perfect square under the square root?” 5 To rationalize an expression like 3 we must ask ourselves, “By what do I 22 3 multiply 12 to get a perfect cube under the cube root?” The perfect cube we want is 3 3 3 23 (since we began with 2) so that 12  222  223  2. We will practice some fill-in-the-blank problems to eliminate radicals before we move on to rationalizing.

Example 4 In-Class Example 4 Fill in the blank. Assume the variable represents positive real numbers. 3 3 3 3 a) 1 6 2 ? 2 6 6 3 3 3 3 b) 2 49  2 ? 2 7 7 3 3 3 c) 2?  2m  2m3  m 4 4 4 4 d) 2 4 2 ? 2 2 2 5 5 5 5 e) 2 4 2 ? 2 2 2 3 2 3 answer: a) 26 b) 2 7 3 4 2 5 3 2 c) 2m d) 22 e) 2 2

Fill in the blank. Assume the variable represents positive real numbers. 3 3 3 3 3 3 a) 15  1?  253  5 b) 23  1?  233  3 3 3 3 5 5 5 c) 2x2  1?  2x3  x d) 18  1?  225  2 4 4 4 e) 127  1?  234  3

Solution 3 3 3 a) Ask yourself, “By what do I multiply 25 to get 253?” The answer is 252. 3 3 3 25  1?  253  5 3 2 3 3 25  25  2 5 5 3

3 3 3 b) “By what do I multiply 13 to get 233?” 232 3 3 3 13  1?  233  3 3 3 2 3 3 1 3 2 3 2 3 3 3 3 3 c) “By what do I multiply 2x2 to get 2x3?” 1x 3 3 3 2x2  1?  2x3  x 3 2 3 3 3 2x  1x  2x  x 5 5 5 5 d) In this example, 18  1?  225  2, why are we trying to obtain 225 instead 5 5 5 of 28 ? Because in the first radical, 18, 8 is a power of 2. Before attempting to fill in the ?, rewrite 8 as 23. 5 5 5 1 8  1?  225  2 5 5 5 223  1?  225  2 5 3 5 2 5 5 22  22  22  2

e) Fill in the blank for this problem using the same approach as the problem above. 4 4 Since 27 is a power of 3, rewrite 127 as 233. 4 4 4 127  1?  234  3 4 4 4 233  1?  234  3 4 3 4 4 23  13  234  3

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Chapter 10 Radicals and Rational Exponents

You Try 4 Fill in the blank. Assume the variable represents a positive real number. 3 3 3 a) 2 2  2?  223  2

5 2 5 5 b) 2 t  2?  2t5  t

4 4 4 c) 2 125  2?  254  5

We use the technique presented in Example 4 to rationalize denominators with indices higher than 2.

Example 5 In-Class Example 5 Rationalize the denominator. Assume the variable represents a positive real numbers. 8 9 4 2 a) 3 b) c) 5 A 9 1t 113 3 82 9 answer: a) 3 5 92t4 c) t

4 2 18 b) 3

Rationalize the denominator. Assume the variable represents a positive real number. 7 3 7 a) 3 b) 5 c) 4 A4 13 2n

Solution a) To rationalize the denominator of

7

, first identify what we want to get as the 13 3 3 3 3 radicand after multiplying. We want to obtain 2 3 since2 3  3. 7 3

13

 ___ 

3

d This is what we want to get.

3 3 2 3

c What is needed here? 3 3 3 3 2 Ask yourself, “By what do I multiply 1 3 to get 2 3 ?” 2 3

7 3 1 3



3 2 2 3



3 2 2 3

3 2 72 3

Simplify.

b) Use the quotient rule for radicals to rewrite 5 2 3 5 2 4

Working with

Multiply.

3 3 2 3 3 71 9  3

5 3 1 3 as 5 . Then, write 4 as 22 to get A4 14 5

5



23 5 2 2 2

5 2 3

, identify what we want to get as the radicand in the 5 2 2 2 5 5 denominator after multiplying. We want to obtain 225 since 225  2. 5 2 3 5 2 2 2

 ___ 

5 5 2 2

c What is needed here?

d This is what we want to get.

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5 2 5 5 5 3 “By what do I multiply 2 2 to get 2 2 ?” 2 2 5 2 3 5 2 2 2



5 3 2 2 5 3 2 2



5 5 3 2 3 2 2

5 5 2 2 5 5 13  1 8  2 5 1 24  2

Multiply.

Multiply.

7

c) To rationalize the denominator of

, first identify what we want to get as the 1n 4 4 4 4 radicand after multiplying. We want to obtain 2 n since 2 n  n. 7 4

2n

 ___ 

4

d This is what we want to get.

4 4 2 n

c What is needed here? 4 4 4 4 3 Ask yourself, “By what do I multiply 2 n to get 2 n ?” 2 n

7 4 2 n



4 3 2 n 4 3 2 n



4 3 72 n

4 4 2 n 4 3 72 n  n

Multiply. Simplify.

You Try 5 Rationalize the denominator. Assume the variable represents a positive number. a)

4 3

17

b)

4 2 B 27

c)

5 8 B w3

3. Rationalize a Denominator Containing Two Terms 1 , we multiply the numer5  13 ator and the denominator of the expression by the conjugate of 5  13. To rationalize the denominator of an expression like

Definition

The conjugate of a binomial is the binomial obtained by changing the sign between the two terms.

Expression

Conjugate

5  13 17  215 1a  1b

5  13 17  215 1a  1b

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Chapter 10 Radicals and Rational Exponents

In Section 10.5 we applied the formula (a  b)(a  b)  a2  b2 to multiplying binomials containing square roots. Recall that the terms containing the square roots were eliminated.

Example 6 In-Class Example 6 Multiply 4  13 by its conjugate. answer: 13

Multiply 8  16 by its conjugate.

Solution The conjugate of 8  16 is 8  16. We will first multiply using FOIL to show why the radical drops out, then we will multiply using the formula (a  b)(a  b)  a2  b2. i) Use FOIL to multiply. (8  16)(8  16)  8  8  8  16  8  16  16  16 F O I L  64  6  58 ii) Use (a  b)(a  b)  a2  b2. (8  16)(8  16)  (8) 2  ( 16) 2  64  6  58

Substitute 8 for a and 16 for b.

Each method gives the same result.

You Try 6 Multiply 2  111 by its conjugate.

To rationalize the denominator of an expression in which the denominator contains two terms, multiply the numerator and denominator of the expression by the conjugate of the denominator.

Example 7

Rationalize the denominator and simplify completely. Assume the variables represent positive real numbers. 3 1a  b a) b) 5  13 1b  a

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In-Class Example 7 Rationalize the denominator and simplify completely. Assume the variables represent positive real numbers. 3 1c  d a) b) 6  12 1d  c 18  312 answer: a) 34 1cd  c 1c  d1d  cd b) d  c2

681

Solution 3 has two terms. To rationalize the 5  13 denominator we must multiply the numerator and denominator by 5  13, the conjugate of the denominator.

a) First, notice that the denominator of

3 5  13  5  13 5  13 3(5  13)  (5) 2  ( 13) 2 15  3 13  25  3 15  3 13  22

Multiply by the conjugate. (a  b)(a  b)  a2  b2 Square the terms. Subtract.

b) The conjugate of 1b  a is 1b  a. 1a  b 1b  a  1b  a 1b  a

Multiply by the conjugate.

In the numerator we must multiply ( 1a  b)( 1b  a). Since these are binomials, we will use FOIL. 1ab  a 1a  b 1b  ab ( 1b) 2  (a) 2 1ab  a 1a  b 1b  ab  b  a2 

FOIL (a  b)(a  b)  a2  b2 Square the terms.

You Try 7 Rationalize the denominator and simplify completely.Assume the variables represent positive real numbers. a)

1 17  2

b)

c  1d c  1d

4. Simplify a Radical Expression by Dividing Out Common Factors From the Numerator and Denominator Sometimes it is necessary to simplify a radical expression by dividing out common factors from the numerator and denominator. This is a skill we will need in Chapter 11 when we are solving quadratic equations, so we will look at an example here.

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Chapter 10 Radicals and Rational Exponents

Example 8 Simplify completely: In-Class Example 8 Simplify completely 913  18 9 answer: 13  2

4 15  12 4

Solution

It is tempting to do one of the following: 415  12  15  12 4 or 3

4 15  12  415  3 4 Each is incorrect because 415 is a term in a sum and 12 is a term in a sum.

4 15  12 is to begin by factoring out a 4 in the 4 numerator and then divide the numerator and denominator by any common factors. The correct way to simplify

4( 15  3) 4 15  12  4 4

Factor out 4 from the numerator.

1

4 ( 15  3)  4

Divide by 4.

 15  3

Simplify.

1

4( 15  3) because the 4 in the 4 numerator is part of a product not a sum or difference. We can divide numerator and denominator by 4 in

You Try 8 Simplify completely. a)

5 17  40 5

b)

20  612 4

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Answers to You Try Exercises 5 1p 3k2 15km 17 513 9130 114 115 b) c) 2) a) b) 3) a) b) 7 3 5 7 5 p 5m 3 4 5 2r 15rs 41 49 1 6 2 8w 2 2 3 c) 4) a) 2 or 4 b) t c) 5 5) a) b) c) 6) 7 s 7 3 w 2 17  2 c  2c1d  d 10  3 12 7) a) b) 8) a) 17  8 b) 2 3 2 c d 1) a)

10.7 Exercises

 Practice  Problems

 NetTutor  Self-Test

 e-Professors

 Videos

Objective 1

Simplify completely. Assume all variables represent positive real numbers.

1) What does it mean to rationalize the denominator of a radical expression? Eliminate the radical from the denominator.

2) In your own words, explain how to rationalize the denominator of an expression containing one term in Answers will vary. the denominator.

25)

8 1y

8 1y

27)

15 1t

15t t

10f 3 B g

f110fg

29) 31)

64v7 B 5w

33)

a3b3 B 3ab4

Rationalize the denominator of each expression. 3) 5) 7) video

1 15 3 12 9 16

20 9)  18 13 11) 128

15 5

4)

312 2

6)

316 2

8)

1 16 5 13 25 110

16 6 513 3 5110 2

5 12

18 10)  145

6 15  5

121 14

18 12) 127

216 9

13)

20 A 60

13 3

14)

12 A 80

115 10

15)

18 A 26

3113 13

16)

42 A 35

130 5

142 6

166 18) 112

122 2

156 17) 148

video

37)

19)

10 7  A 7 A3

1 6  21) A5 A8 23)

6 7  A7 A3

41w w

28)

12 1m

12m m

30)

12s6 B r

2s3 13r r

8v3 15vw 5w

32)

81c5 B 2d

9c2 12cd 2d

a 13b 3b

34)

m2n5 B 7m3n

n2 17m 7m

g

175



2b

3

5 13b b

113

113j

2j

j3

5

36) 

2

38)

20)

115 10

4 1  22) A6 A5 24)

11 5  A 5 A2

11 8  A 10 A 11

124 2v

3



216v v2

122

122w

2w

w4

7

Fill in the blank. Assume all variables represent positive real numbers. 3 3 3 39) 12  1?  223  2

22 or 4

3 3 3 40) 15  1?  253  5

52 or 25

3 3 3 41) 19  1?  233  3

3

3

3

3

3

3

3

3

3

42) 14  1?  22  2

130 3

12

4 1w

Objective 2

Multiply and simplify. video

35) 

26)

y

122 2 130 15 2 15 5

2

43) 1c  1?  2c  c

c2

3 3 3 44) 1p  1?  2p3  p

p2

5 5 5 45) 14  1?  225  2

23 or 8

5 5 5 46) 116  1?  225  2

2

4

4

4

47) 2m  1?  2m  m 4

3

4

4

4

48) 1k  1?  2k4  k

m k3

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Chapter 10 Radicals and Rational Exponents

Rationalize the denominator of each expression. Assume all variables represent positive real numbers. video

49) 51) 53)

3

13 12 3

12 9 3

125

3 41 9 3

50)

3 61 4

52)

3

915 5

54)

4

26 3

15 21

3 26 1 25 5

81)

3 6  3 13 2  13

82)

8 6  15

48  815 31

3 71 9

83)

90  1012 10 79 9  12

84)

5 4  16

4  16 2

3 31 2

85)

18 2 16  4 13  12

86)

132 2 110  2114 15  17

87)

13  15 110  13

88)

13  16 12  15

89)

m  1mn 1m 1m  1n m  n

90)

1u 1u  1v

u  1uv uv

b  25 1b  5 1b  5

92)

d9 1d  3

1d  3

3

13 6 3

14

55)

5 A9

145 3

56)

2 A 25

4 1 50 5

57)

3 A8

5 1 12 2

58)

7 A4

5 1 56 2

59)

2 A9

4 1 18 3

60)

10 A 27

4 1 30 3

91)

3 2 102 z z

62)

1u

3 2 62 u u

93)

3 1 3n n

64)

5 A x2

3 1 5x x

61) 63) 65) 67)

video

4

Rationalize the denominator and simplify completely. Assume the variables represent positive real numbers.

4

5

4

10 3

1z 3 A n2 3

3 1 7 3 2 2k2

9 5

2a3

3 1 28k 2k

66)

4

5

4

6 3

3

3 1 2 3 1 25t

5

92a2 a

68)

8 5

2h

2

18  617 6

3  17

Objective 4

Simplify completely.

36  2012 30  1815 15  9 15 98) 2 4 12

99)

145  6 9

4 2 3 2 xy

71)

5 A 2m

4 2 40m3 2m

72)

2 A 3t2

4 2 54t2 3t

video

y

101)

10  150 5

2  12

Objective 3

73) How do you find the conjugate of a binomial? Change the sign between the two terms.

74) When you multiply a binomial containing a square root by its conjugate, what happens to the radical?

Find the conjugate of each binomial. Then, multiply the binomial by its conjugate. 75) (5  12) (5  12); 23 76) ( 15  4) ( 15  4); 11 77) ( 12  16)

78) ( 13  110) ( 13  110 ); 7

79) ( 1t  8)

80) ( 1p  5) ( 1p  5); p  25

(1t  8); t  64

fg

97)

x2 By

( 12  16 ); 4

1g  1f

5 3 82 h h

70)

4

1x  1y

1f  1g f  21fg  g

5  1013 5

4 1 cd d

4

94)

95)

c A d3

4

1x  1y

3 2 10t2 5t

69)

4

video

1  2 13

15  2 3

96)

9  5 12 3

100)

148  28 4

102)

35  1200 7  212 3 15

13  7

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Section 10.8 Solving Radical Equations Objectives 1. Understand the Steps Used to Solve a Radical Equation 2. Solve an Equation Containing One Radical Expression 3. Solve an Equation Containing Two Radicals That Does Not Require Squaring a Binomial 4. Square a Binomial Containing a Radical Expression 5. Solve an Equation Containing Two Radicals That Requires Squaring a Binomial

In this section, we will discuss how to solve radical equations. An equation containing a variable in the radicand is a radical equation. Some examples of radical equations are 1c  3

3 1 n2

12x  1  1  x

15w  6  14w  1  1

1. Understand the Steps Used to Solve a Radical Equation Before we present the steps for solving radical equations, we need to answer two important questions which may arise from those steps. 1) How do we eliminate the radical? 2) Why is it absolutely necessary to check the proposed solution(s) in the original equation? 1) How do we eliminate the radical? To understand how to eliminate a radical, we will revisit the relationship between roots and powers. ( 1x) 2  x since ( 1x) 2  (x12 ) 2  x 3 3 (1 x) 3  x since ( 1 x) 3  (x13 ) 3  x 4 4 4 ( 1x)  x since ( 1x) 4  (x14 ) 4  x To eliminate a radical from expressions like those above, raise the expression to the power equal to the index. We must also keep in mind when solving an equation that whatever operation is done to one side of the equation must be done to the other side as well. 2) Why is it absolutely necessary to check the proposed solution(s) in the original equation? There may be extraneous solutions. In the process of solving an equation we obtain new equations. An extraneous solution is a value that satisfies one of the new equations but does not satisfy the original equation. Extraneous solutions occur frequently when solving radical equations, so we must check all “solutions” in the original equation and discard those that are extraneous.

Steps for Solving Radical Equations 1)

Get a radical on a side by itself.

2)

Eliminate a radical by raising both sides of the equation to the power equal to the index of the radical.

3)

Combine like terms on each side of the equation.

4)

If the equation still contains a radical, repeat steps 1–3.

5)

Solve the equation.

6)

Check the proposed solutions in the original equation and discard extraneous solutions.

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Chapter 10 Radicals and Rational Exponents

2. Solve an Equation Containing One Radical Expression Example 1 In-Class Example 1 Solve. a) 1b  36 3 b) 1 x  125 c) 1w  3  3  0 answer: a) 6 b) 5

c) 

Solve. a) 1c  3

3 b) 1n  2

1t  5  6  0

c)

Solution a) Step 1: The radical is on a side by itself. 1c  3 Step 2: Square both sides to eliminate the square root. ( 1c) 2  32 c9

Square both sides.

Steps 3 and 4 do not apply because there are no like terms to combine and no radicals remain. Step 5: The solution obtained above is c  9. Step 6: Check c  9 in the original equation. 1c  3 ? 19  3

1

The solution set is {9}. b) Step 1: The radical is on a side by itself. 3 1 n2

Step 2: Cube both sides to eliminate the cube root. 3 (1 n) 3  23 n8

Cube both sides.

Steps 3 and 4 do not apply because there are no like terms to combine and no radicals remain. Step 5: The solution obtained is n  8. Step 6: Check n  8 in the original equation. 3 1 n2 ? 3 1 82

1

The solution set is {8}. c) Step 1: Get the radical on a side by itself. 1t  5  6  0 1t  5  6

Subtract 6 from each side.

Step 2: Square both sides to eliminate the square root. ( 1t  5) 2  (6) 2 t  5  36

Square both sides. The square root is eliminated.

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Steps 3 and 4 do not apply because there are no like terms to combine on each side and no radicals remain. Step 5: Solve the equation. t  5  36 t  31 Step 6:

Subtract 5 from each side.

Check t  31 in the original equation. 1t  5  6  0 ? 131  5  6  0 ? 136  6  0 6  6  0 False

t  31 does not satisfy the original equation. It is an extraneous solution. There is no real solution to this equation. The solution is .

You Try 1 Solve. a) 1a  9

Example 2 In-Class Example 2 Solve 12y  4  2  y. answer: {2, 0}

3 b) 1 y34

c) 1m  7  3  0

Solve 12x  1  1  x.

Solution Step 1: Get the radical on a side by itself. 12x  1  1  x 12x  1  x  1

Subtract 1 from each side.

Step 2: Square both sides to eliminate the square root. ( 12x  1) 2  (x  1) 2

Square both sides.

Use FOIL or the formula (a  b) 2  a2  2ab  b2 to square the right side! 2x  1  x2  2x  1 Steps 3 and 4 do not apply because there are no like terms to combine on each side and no radicals remain. Step 5: Solve the equation. 2x  1  x2  2x  1 0  x2  4x 0  x(x  4) x  0 or x  0 or

x40 x4

Subtract 2x and subtract 1. Factor. Set each factor equal to zero. Solve.

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Chapter 10 Radicals and Rational Exponents

Step 6: Check x  0 and x  4 in the original equation. x  0:

12x  1  1  x ? 12(0)  1  1  0 ? 11  1  0 ? 20 False

x  4:

12x  1  1  x ? 12(4)  1  1  4 ? 18  1  1  4 ? 19  1  4 3  1  4 True

x  4 is a solution but x  0 is not because x  0 does not satisfy the original equation. The solution set is {4}.

You Try 2 Solve. a) 13p  10  4  p

b) 14h  3  h  2

3. Solve an Equation Containing Two Radicals That Does Not Require Squaring a Binomial Now we turn our attention to solving equations containing two radicals. We use the same steps presented earlier, but in some cases it will be necessary to repeat steps 1–3 if a radical remains after performing these steps once.

Example 3 In-Class Example 3 Solve 13b  4  2 1b  1. answer: {8}

3 3 Solve 1 7a  1  2 1 a  1  0.

Solution Step 1: Get a radical on a side by itself. 3 3 1 7a  1  2 1 a3

3 Add 21 a  1.

Step 2: Cube both sides to eliminate the cube roots. 3 3 (1 7a  1) 3  (21 a  3) 3 7a  1  8(a  1)

Cube both sides. 23  8

Step 3 and 4 do not apply because there are no like terms to combine on each side and no radicals remain. Step 5: Solve the equation. 7a  1  8(a  1) 7a  1  8a  8 1a8 9a

Step 6: Verify that a  9 satisfies the original equation. The solution set is {9}.

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689

You Try 3 3

3

Solve 3 1r  4  1 5r  2  0.

4. Square a Binomial Containing a Radical Expression Recall that to square a binomial we can either use FOIL or one of the formulas (a  b) 2  a2  2ab  b2 (a  b) 2  a2  2ab  b2

Example 4 In-Class Example 4 Square the binomial and simplify (6  1r  1) 2. answer: 37  121r  1  r

Square the binomial and simplify (3  2m  2) 2.

Solution Use the formula (a  b) 2  a2  2ab  b2. (3  1m  2) 2  (3) 2  2(3)( 1m  2)  ( 1m  2) 2  9  6 1m  2  (m  2)  m  11  61m  2

Substitute 3 for a and 1m  2 for b. Combine like terms.

You Try 4 Square each binomial and simplify. a) ( 1z  4) 2

b) (5  13d  1) 2

5. Solve an Equation Containing Two Radicals That Requires Squaring a Binomial Example 5 In-Class Example 5 Solve each equation. a) 1c  1c  9  1 b) 17k  8  13k  4  2 answer: a) 25 b) 4

Solve each equation. a) 1x  1x  7  1

b) 15w  6  14w  1  1

Solution a) Step 1: Get a radical on a side by itself. 1x  1x  7  1 1x  1  1x  7

Add 1x  7 to each side.

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Chapter 10 Radicals and Rational Exponents

Step 2: Square both sides of the equation to eliminate a radical. ( 1x) 2  (1  1x  7) 2 x  (1) 2  2(1)( 1x  7)  ( 1x  7) 2 x  1  2 1x  7  x  7

Square both sides. Use the formula (a  b) 2  a2  2ab  b2.

Step 3: Combine like terms on the right side. x  x  6  2 1x  7

Combine like terms.

Step 4: The equation still contains a radical, so repeat steps 1–3. Step 1: Get the radical on a side by itself. x  x  6  2 1x  7 0  6  2 1x  7 6  2 1x  7 3  1x  7

Subtract x from each side. Add 6 to each side. Divide both sides by 2 to make the numbers smaller to work with.

Step 2: Square both sides to eliminate the square root. 32  ( 1x  7) 2 9x7

Square both sides.

Steps 3 and 4 don’t apply. Step 5: Solve the equation. 9x7 16  x

Step 6: Verify that x  16 satisfies the original equation. The solution set is {16}. b) Step 1: Get a radical on a side by itself. 15w  6  14w  1  1 15w  6  1  14w  1

Add 14w  1 to each side.

Step 2: Square both sides of the equation to eliminate a radical. Square both sides. ( 15w  6) 2  (1  14w  1) 2 5w  6  (1) 2  2(1)( 14w  1)  ( 14w  1) 2 Use the formula (a  b) 2 

5w  6  1  2 14w  1  4w  1

a2  2ab  b2.

Step 3: Combine like terms on the right side. 5w  6  4w  2  214w  1

Combine like terms.

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691

Step 4: The equation still contains a radical, so repeat steps 1–3. Step 1: Get the radical on a side by itself. 5w  6  4w  2  2 14w  1 w  6  2  2 14w  1 w  4  2 14w  1

Subtract 4w. Subtract 2.

We do not need to eliminate the 2 from in front of the radical before squaring both sides. The radical must not be a part of a sum or difference when we square. Step 2: Square both sides of the equation to eliminate the radical. (w  4) 2  (2 14w  1) 2 w  8w  16  4(4w  1)

Square both sides.

2

On the left, we squared the binomial using the formula (a  b) 2  a2  2ab  b2. On the right, don’t forget to square the 2 in front of the square root. Steps 3 and 4 do not apply. Step 5: Solve the equation. w2  8w  16 w2  8w  16 w2  8w  12 (w  6)(w  2)

 4(4w  1)  16w  4 0 0

w  6  0 or w  6 or

w20 w2

Distribute. Get all terms on the left. Factor. Set each factor equal to zero. Solve.

Step 6: Verify that w  6 and w  2 each satisfy the original equation. The solution set is {2, 6}.

Watch out for two common mistakes that students make when solving an equation like the one in (Example 5b.) 1)

Do not square both sides before getting a radical on a side by itself. This is incorrect: ( 15w  6  14w  1) 2  12 5w  6  (4w  1)  1

2)

The second time we perform step 2, watch out for this common error: This is incorrect: (w  4) 2  (214w  1) 2 w 2  16  2(4w  1) On the left we must multiply using FOIL or the formula (a  b)2  a2  2ab  b2 and on the right we must remember to square the 2.

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Chapter 10 Radicals and Rational Exponents

You Try 5 Solve each equation. a) 12y  1  1y  1

b) 13t  4  1t  2  2

Using Technology In Chapter 9 we learned how to solve an absolute value equation using a graphing calculator. Now we will see how the algebraic solution of an equation containing a radical is related to graphs. In Example 2 of this section we solved 12x  1  1  x. The proposed solutions we obtained were 0 and 4, but when we checked these values in the original equation, only 4 satisfied the equation. In a graphing calculator we can enter the left side of the equation as Y1 and the right side as Y2. Press GRAPH to see the graphs of each equation on the same screen. 10

−10

10

−10 Notice that the graphs intersect at only one point: (4, 4). Recall that the solution to the equation is the x-coordinate of the point of intersection. We obtain the same result that was obtained by solving the equation algebraically.The solution set is {4}. Use a graphing calculator to verify the results of Examples 1, 3, and 5. Solve each equation. 1.

1c  3 (Example 1a)

2.

3 1 n  2 (Example 1b)

3.

1t  5  6  0 (Example 1c)

4.

3 3 1 7a  1  2 1 a  1  0 (Example 3)

5.

1x  1x  7  1 (Example 5a)

6.

15w  6  14w  1  1 (Example 5b)

Answers to You Try Exercises 1) a) {81} b) {61} c)  2) a) {3, 2} b) {7} 3) {5} 4) a) z  81z  16 b) 3d  24  1013d  1 5) a) {0, 4}

b) {1}

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10.8 Exercises

 Practice  Problems

 NetTutor  Self-Test

 e-Professors

693

 Videos

40) 3  18t  9  t

{0, 2}

41) 3v  8  13v  4

{4}

Sometimes there are extraneous solutions.

42) 4k  3  110k  5

{2}

Raise both sides of the equation to the third power.

Objective 4

Objectives 1–3

1) Why is it necessary to check the proposed solutions to a radical equation in the original equation? 2) How do you eliminate the radical from an equation 3 like 1 x  2?

Use (a  b)2  a2  2ab  b2 or (a  b)2  a2  2ab  b2 to multiply each of the following binomials.

Solve. 3) 1q  7

{49}

2 5) 1w   0 3

4 e f 9

3 11) 1 m  4

video

{64}

{100}

3 6) 1r   0 5

9 e f 25

43) 1 1x  52 2 44) 1 1y  82

3 10) 1 c  3

{27}

3 12) 1 t  2

{8}

46) 14  1p  52

y  161y  64 2

85  181a  4  a

2

21  81p  5  p

47) 1213n  1  72 2

13) 1b  11  3  0 {20} 14) 1d  3  5  0 {22}

48) 15  312k  32

15) 14g  1  7  1  16) 13l  4  10  6

Objective 5

 2 29 e f 17) 13f  2  9  11 3 18) 15u  4  12  17 e f 5 3 3 3 19) 1 2x  5  3  1 e f 20) 1 4a  1  7  4 {7} 2

21) 12c  3  15c video

x  101x  25

2

45) 19  1a  42

8) 1k  8  2 

7) 1a  5  3  3 9) 1 y  5 {125}

4) 1z  10

{1} 3

3

23) 1 6j  2  1 j  7 {1}

22) 15w  2  14w  2 3

3

{4}

24) 1 m  3  1 2m  11

{14}

2

12n  28 13n  1  45 18k  3012k  3  2

Solve. 49) 12y  1  2  1y  4

{5, 13}

50) 13n  4  12n  1  1

{0, 4}

51) 1  13s  2  12s  5

{2}

25) 5 11  5h  4 11  8h

{3}

52) 14p  12  1  16p  11

{6}

26) 3 16a  2  4 13a  3

{11}

53) 13k  1  1k  1  2

{1, 5}

27) 3 13x  6  2 19x  9

{10}

54) 14z  3  15z  1  1

{3, 7}

28) 5 1q  11  2 18q  25

{25}

55) 13x  4  5  13x  11



29) Multiply (x  3) 2.

x2  6x  9

56) 14c  7  14c  1  4



30) Multiply (2y  5) 2.

4y2  20y  25

57) 13v  3  1v  2  3

{2, 11}

58) 12y  1  1y  1

{0, 4} 1 e f 4 {3}

Solve.

video

59) 15a  19  1a  12  1

31) m  2m2  3m  6

{2}

32) b  2b2  4b  24

{6}

33) p  6  112  p

{3}

Solve.

34) c  7  12c  1

{12}

61) 213  1r  1r  7

{9}

35) 2r2  8r  19  r  9

{10}

62) 1m  1  2m  1m  4

{5}

36) 2x2  x  4  x  8

{4}

63) 2y  1y  5  1y  2

{1}

37) 6  2c2  3c  9  c



64) 22d  1d  6  1d  6

{10}

38) 4  2z2  5z  8  z



39) w  110w  6  3

{1, 3}

60) 12u  3  15u  1  1

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Chapter 10 Radicals and Rational Exponents video

Solve for the indicated variable. 65) v 

2E 300VP mv for E E  66) V  for P mV 2 Am A m P 2

73) Let V represent the volume of a cylinder, h represent its height, and r represent its radius. V, h, and r are related according to the formula

300

67) c  2a  b for b 2

b2  c2  a2

video

2

2

A 68) r  for A Ap

r

A  pr 2

E 4 4 E 3 3V 69) T  for s s  4 70) r  for V V  pr3 3 T As A 4p

71) The speed of sound is proportional to the square root of the air temperature in still air. The speed of sound is given by the formula

V A ph

a)

A cylindrical soup can has a volume of 28 in3. It 2 in. is 7 in. high. What is the radius of the can?

b)

Solve the equation for V.

V  r 2h

74) For shallow water waves the wave velocity is given by c  1gH

VS  201T  273 where VS is the speed of sound in meters/second and T is the temperature of the air in °Celsius. a)

What is the speed of sound when the temperature is 17°C (about 1°F2? 320 m/sec

b)

What is the speed of sound when the temperature is 16°C (about 61°F2? 340 m/sec

c)

What happens to the speed of sound as the temperature increases? The speed of sound increases.

d)

Solve the equation for T.

T

vs2  273 400

72) If the area of a square is A and each side has length l, then the length of a side is given by l  1A. A square rug has an area of 25 ft2. a)

Find the dimensions of the rug.

5 ft  5 ft

b)

Solve the equation for A.

A  l2

where g is the acceleration due to gravity (32 ft/sec2 ) and H is the depth of the water (in feet). a)

Find the velocity of a wave in 8 ft of water. 16 ft/sec

b)

Solve the equation for H.

H

c2 g

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Chapter 10: Summary Definition/Procedure

Example

Reference

Finding Roots

10.1

If the radicand is a perfect square, then the square root is a rational number.

149  7 since 72  49

p. •••

If the radicand is a negative number, then the square root is not a real number.

236 is not a real number.

p. •••

If the radicand is positive and not a perfect square, then the square root is an irrational number.

27 is irrational because 7 is not a perfect square.

p. •••

The 1 a is read as “the nth root of a.” If 1 a  b, then bn  a. n is the index of the radical.

5 1 32  2 since 25  32

p. •••

The odd root of a negative number is a negative number.

3 1 125  5 since (5) 3  125

p. •••

The even root of a negative number is not a real number.

1 16 is not a real number.

n

n

p. •••

4

Rational Exponents

10.2

If a is a nonnegative number and n is a positive n integer, then a1n  1 a.

3 813  1 8  2.

p. •••

If a is a nonnegative number and m and n are integers such that n is positive, then n amn  (a1n ) m  ( 1 a) m

4 1634  ( 1 16) 3  23  8

p. •••

If a is positive number and m and n are integers such that n is positive, then 1 mn 1 amn  a b  mn . a a

2532  a

1 32 1 3 1 3 1 b a b  b a B 25 25 5 125

p. •••

The negative exponent does not make the expression negative.

If a is a nonnegative real number and n is a posin n tive integer, then ( 1 a) n  a and 1 an  a.

( 119) 2  19 4 4 2 t  t (provided t represents a nonnegative real number)

Simplifying Expressions Containing Square Roots Product Rule for Square Roots

p. •••

10.3 15  17  25  7  135

p. •••

Let a and b be nonnegative real numbers.Then, 1a  1b  1ab

Chapter 10 Summary

695

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Definition/Procedure

Example

Reference p. •••

An expression containing a square root is simplified when all of the following conditions are met: 1) The radicand does not contain any factors (other than 1) which are perfect squares. 2) The radicand does not contain any fractions. 3) There are no radicals in the denominator of a fraction.

Simplify 124. 124  14  6  14  16  2 16

4 is a perfect square. Product rule 14  2

To simplify square roots, reverse the process of multiplying radicals, where a or b is a perfect square. 1ab  1a  1b After simplifying a radical, look at the result and ask yourself, “Is the radical in simplest form?” If it is not, simplify again. Quotient Rule for Square Roots Let a and b be nonnegative real numbers a 2a such that b  0. Then,  Bb 2b

If a is a nonnegative real number and m is an integer, then 2am  am2.

72 272  B 25 225 236  22  5 622  5

p. ••• Quotient rule Product rule; 125  5 136  6

2k18  k182  k9 (provided k represents a nonnegative real number)

p. •••

Two Approaches to Simplifying Radical Expressions Containing Variables Let a represent a nonnegative real number.To simplify 2an where n is odd and positive, i) Method 1: Write an as the product of two factors so that the exponent of one of the factors is the largest number less than n that is divisible by 2 (the index of the radical). ii) Method 2: 1) Divide the exponent in the radicand by the index of the radical. 2) The exponent on the variable outside of the radical will be the quotient of the division problem. 3) The exponent on the variable inside of the radical will be the remainder of the division problem.

696

Chapter 10 Radicals and Rational Exponents

p. •••

i) Simplify 2x9. 2x9  2x8  x1  2x8  1x  x82 1x  x4 1x ii) Simplify 2p15. 2p15  p7 2p1 7  p1 p

8 is the largest number less than 9 that is divisible by 2. Product rule 824 15  2 gives a quotient of 7 and a remainder of 1.

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Definition/Procedure

Example

Reference

Simplifying Expressions Containings Higher Roots Product Rule for Higher Roots

10.4 3 3 3 1 3 1 5 1 15

p. •••

3 Simplify 1 40.

p. •••

If a and b are real numbers such that the roots n n n exist, then 1 a  1 b  1 a  b Let P be an expression and let n be a positive real n number.Then 1 P is completely simplified when all of the following conditions are met: 1) The radicand does not contain any factors (other than 1) that are perfect nth powers. 2) The radicand does not contain any fractions. 3) There are no radicals in the denominator of a fraction. To simplify radicals with any index, reverse the process of multiplying radicals where a or b is an nth power.

Method 1: i) Think of two numbers that multiply to 40 so that one of them is a perfect cube. 40  8  5

8 is a perfect cube.

3 3 Then, 1 40  1 85 3 3  18  1 5 3  21 5

Product rule 3 1 82

Method 2: ii) Begin by using a factor tree to find the prime factorization of 40. 40  23  5 3 3 1 40  2 2 5 3 3 3 2 2 1 5 3  21 5 3

Quotient Rule for Higher Roots If a and b are real numbers such that the roots n a 1a exist and b  0, then n  n Ab 1b Simplifying Higher Roots with Variables in the Radicand If a is a nonnegative number and m and n are inten gers such that n is positive, then 2am  amn. If the exponent does not divide evenly by the index, we can use two methods for simplifying the radical expression. If a is a nonnegative number and m and n are integers such that n is positive, then i) Method 1: Use the Product rule: n To simplify 2am, write am as the product of two factors so that the exponent of one of the factors is the largest number less than m that is divisible by n (the index).

Product rule 3 3 2 2 2

4 4 4 4 32 1 32 1 16  1 2 21 2  4   A 81 3 3 1 81

p. •••

4 12 Simplify 2 a .

p. •••

4

4 12 2 a  a124  a3

5 17 i) Simplify 2 c . 5 17 5 15 2 c 2 c  c2 5 15 5 2 2 c 2 c 5 2 155 c  2c 5 2  c3 2 c 4

15 is the largest number less than 17 that is divisible by 5. Product rule

p. •••

15  5  3

ii) Simplify 2m

11

4 4 2 m11  m2 2m3

11  4 gives a quotient of 2 and a remainder of 3.

ii) Method 2: Use the quotient and remainder (presented in Section 10.3).

Chapter 10 Summary

697

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Definition/Procedure

Example

Reference

10.5 Perform the operations and simplify. a) 5 12  917  312  4 17  212  13 17 b) 172  118  145  136  12  19  12  19  15  612  3 12  3 15  912  3 15

Combining Multiplication, Addition, and Subtraction of Radicals Multiply expressions containing radicals using the same techniques that are used for multiplying polynomials.

p. •••

10.6 Multiply and simplify. Assume all variables represent nonnegative real numbers.

p. •••

a) 1m( 12m  1n)  1m  12m  1m  1n  22m2  1mn  m12  1mn b) ( 1k  16) ( 1k  12) Since we are multiplying two binomials, multiply using FOIL. ( 1k  16) ( 1k  12)  1k  1k  121k  16  1k  16  12 F O I L  k2  12k  16k  112 Product rule  k2  12k  16k  213 112  2 13

To square a binomial we can either use FOIL or one of the special formulas from Chapter 6: (a  b) 2  a2  2ab  b2 (a  b) 2  a2  2ab  b2 To multiply binomials of the form (a  b)(a  b) use the formula (a  b)(a  b)  a2  b2.

( 17  5) 2  ( 17) 2  2( 17) (5)  (5) 2  7  1017  25  32  1017

p. •••

(3  110) (3  110)  (3) 2  ( 110) 2  9  10  1

p. •••

Dividing Radicals The process of eliminating radicals from the denominator of an expression is called rationalizing the denominator. First, we give examples of rationalizing denominators containing one term.

698

Chapter 10 Radicals and Rational Exponents

10.7 Rationalize the denominator of each expression. Assume all variables represent positive real numbers. 9 9   12 12 5 5 b) 3  3  12 12 a)

12 912  2 12 3 2 3 2 3 2 2 52 2 51 4   3 2 3 3 2 2 2 2 2

p. •••

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Definition/Procedure

Example

Reference

The conjugate of a binomial is the binomial obtained by changing the sign between the two terms.

111  4 conjugate: 111  4 8  15 conjugate: 8  25

p. •••

Rationalizing a Denominator with Two Terms To rationalize the denominator of an expression containing two terms, like 4 , multiply the numerator and 12  3 denominator of the expression by the conjugate of 12  3.

Rationalize the denominator of 4 4 12  3   12  3 12  3 12  3 4( 22  3)  ( 12) 2  (3) 2 4( 12  3)  29 4( 12  3)  7 412  12  7

p. •••

4 . 12  3 Multiply by the conjugate. (a  b)(a  b)  a2  b2 Square the terms. Subtract. Distribute.

Solving Radical Equations Steps for Solving Radical Equations 1) Get a radical on a side by itself. 2) Eliminate a radical by raising both sides of the equation to the power equal to the index of the radical. 3) Combine like terms on each side of the equation. 4) If the equation still contains a radical, repeat steps 1–3. 5) Solve the equation. 6) Check the proposed solutions in the original equation and discard extraneous solutions.

10.8 Solve t  2  12t  1.

p. •••

t  2  22t  1 Get the radical by itself. (t  2) 2  ( 12t  1) 2 Square both sides. t2  4t  4  2t  1 Get all terms on the same side. t2  6t  5  0 Factor. (t  5) (t  1)  0 t  5  0 or t  1  0 t  5 or t1 Check t  5 and t  1 in the original equation. t  5: t  2  22t  1 5  2  12(5)  1 5  2  19 523 True

t  1: t  2  12t  1 1  2  12(1)  1 121 13 False

t  5 is a solution, but t  1 is not because t  1 does not satisfy the original equation. The solution set is {5}.

Chapter 10 Summary

699

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Chapter 10: Review Exercises *Additional answers can be found in the Answers to Exercises appendix. (10.1) Find each root, if possible.

1) 125

5

2) 116

3) 181

9

4)

3 5) 1 64

4

5 6) 1 32

3

7) 11 9) 164

169 B 4

13 2 2

4

8) 181

1

6

not real

not real

3

10) 19  16

6

41) 2274

3

42) ( 117) 2

17

3 43) 273

7

5 44) 2t20

t4

4 45) 2k28

k7

46) 2x10

x5

47) 2w6

w3

(10.3) Simplify completely. Assume all variables represent positive real numbers.

48) 128

12) 152

(10.2)

14) Explain how to eliminate the negative from the exponent in an expression like 912. 1 2 1 12  a b

9

6

16) 3215

2

27 13 b 17) a 125

3 5

16 12 18) a b 49

4 7

19) 1614

2

20) 100013

10

23

21) 125 23) a

25

23

64 b 27

1 9 1 27

12

25) 81

27) 8134 29) a

16 9

23

27 b 1000

100 9

35

22) 32 24) a

23

125 b 64

3

51)

52)

148 1121

4 13 11

53) 2k12

k6

54)

40 A m4

2 110

55) 2x9

x4 1x

56) 2y5

y2 1y

57) 245t2

3t15

58) 280n21

4n10 15n

59) 272x7y13

6x3y6 12xy

60)

m11 B 36n2

m5 1m 6n

61) 15  13

115

62) 16  115

3 110

63) 12  112

2 16

64) 2b7  2b3

b5

28) 100023

65) 211x5  211x8 11x6 1x 66) 25a2b  215a6b4 5a4b2 13b 67)

64 125

2200k21

10k8

22k

5

3 69) 1 16

2 12

31) 367  387

9

32) (1694 ) 18

13

4 71) 1 48

33) (815 ) 10

64

34)

82

1 32

1

36) (2k56 )(3k12 )

72 753  713

37) (64a4b12 ) 56 39) a 700

8

32a103b10 38) a

81c5d 9 14 2c b 16c1d2 3d74

113

4 3

2

tu b 7t7u5

Chapter 10 Radicals and Rational Exponents

68)

263c17

3c4

27c

9

(10.4) Simplify completely. Assume all variables represent positive real numbers.

Simplify completely. Assume the variables represent positive real numbers.The answer should contain only positive exponents.

35)

312 7

25 16

1 100

25 b 16

m2

18 A 49

Perform the indicated operation and simplify. Assume all variables represent positive real numbers.

3

30) a

10110

8

1 13 26) a b 27 32

49) 11000

163 17

Evaluate. 15) 3612

217

50)

13) Explain how to write 823 in radical form.

Take the reciprocal of the base. 9

12

4 40) 2362

not real

Approximate each square root to the nearest tenth and plot it on a number line.

11) 134

Rewrite each radical in exponential form, then simplify. Write the answer in simplest (or radical) form. Assume all variables represent nonnegative real numbers.

6 k13 49t6u4

3

3 70) 1 250

2 13

4

72)

4 24 73) 2 z

z6

5 40 74) 2 p

p8

3 20 75) 2 a

3 2 a6 2 a

5 14 7 76) 2 x y

5 4 2 x2y2 xy

3 77) 216z15

3 2z5 1 2

3 78) 280m17n10

3 2m5n3 2 10m2n

h3 3

80)

79)

h12 B 81 4

81 A3 3

c22 B 32d10 5

3

5 12 3

5 2 c4 2 c

2d2

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Perform the indicated operation and simplify. Assume the variables represent positive real numbers. 3 3 81) 13  17

3 1 21

3 3 82) 125  110

4 4 4 4 83) 2 4t7  2 8t10 2t 12t 84)

3

85) 1n  1n

6

2n

5

86)

3 51 2

107)

21

5 x B x16

x

109)

4 3 2 a

12

2a

3 1 a

5

87) 8 15  3 15 11 15

88) 1125  180 9 15

89) 180  148  120

3 3 3 90) 9172  8 19 10 19

91) 3p 1p  7 2p3 4p 1p 92) 9n1n  4 2n3 5n1n 93) 10d2 18d  32d 22d3 12d2 12d

142  6

95) 3 1k( 120k  12) 6k15  3 12k

96) (5  13)(2  13) 7  3 13

97) ( 12r  51s)(3 1s  412r) 23 12rs  8r  15s 98) (215  4) 2 36  16 15 99) (1  1y  1) 2 100) ( 16  15)( 16  15) 1

3 71 4 2

3 2 2 x

3 2 2 2 xy

3

1y 2 3  13

y 3  13 3

111)

8  24 12 8

1  3 12

112)

148  6 10

213  3 5

14 13 3

2  2 1y  1  y

118k 103) 1n

312kn n

20 102) 16 104)

15 3 1 9

108)

4 3 A 4k2

110)

z4 1z  2

3 51 3 4 2 12k2 2k

1z  2

(10.8) Solve.

114) 10  13r  5  2 {23} 115) 13j  4  14j  1  3 116) 16d  14  2 {1}

117) a  1a  8  6

1016 3

145

325m

2m5

m3

4

118) 1  16m  7  2m

{3}

119) 14a  1  1a  2  3

{2, 6}

120) 16x  9  12x  1  4

{12}

(10.7) Rationalize the denominator of each expression.

14 101) 13

106) 

113) 1x  8  3 {1}

(10.6) Multiply and simplify. Assume all variables represent nonnegative real numbers.

94) 16( 17  16)

7 3 1 2

Simplify completely.

(10.5) Perform the operations and simplify. Assume all variables represent nonnegative real numbers.

615  413

105)

121) Solve for V: r 

3V A ph

1 V  pr2h 3

122) The velocity of a wave in shallow water is given by c  1gH , where g is the acceleration due to gravity (32 ft/sec2 ) and H is the depth of the water (in feet). Find the velocity of a wave in 10 ft of water. 8 15 ft/sec or about 17.9 ft/sec

Chapter 10 Review Exercises

701

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Chapter 10: Test *Additional answers can be found in the Answers to Exercises appendix. Find each root, if possible. 3

Multiply and simplify. Assume all variables represent nonnegative real numbers.

1) 1144

12

2) 1125

5

3

3

4) 116



3) 127 Evaluate.

5) 161/4 7) 1492

6) 274/3

2

2/3

8 8) a b 125

1 7

1/2

m5/8

11) (2x3/10y2/5 ) 5

10) y2

213  5 16

28) (3  215)( 12  1)

3 12  3  2 110  2 15

29) ( 17  13)( 17  13)

4

81

30) ( 12p  1  2) 2

2p  5  4 12p  1

25 4

31) 21t( 1t  13u)

2t  213tu

Simplify completely. Assume the variables represent positive numbers.The answer should contain only positive exponents.

9) m3/8  m1/4

27) 16( 12  5)

35a1/6

5

14a5/6

2a2/3

32x3/2

Rationalize the denominator of each expression. Assume all variables represent positive real numbers.

32)

2 15

2 15 5

33)

34)

16 1a

16a a

35)

Simplify completely.

12) 175

5 13

24 A2

2 13

14)

3 13) 1 48

36) Simplify completely.

3

2 16

15) 2y6

y3

4 24 16) 2 p

p6

17) 2t9

t4 1t

18) 263m5n8

3m2n4 17m

3 23 19) 2 c

3 2 c7 2 c

20)

a14b7 B 27 3

3 2 a4b2 2 ab 3

Perform the operations and simplify. Assume all variables represent positive real numbers.

23)

6

3 4 3 6 22) 2 z 2 z

3 z3 1 z

2w5 215w

24) 917  3 17

6 27

2120w15 22w4

25) 112  1108  118 322  4 23

702

5 3

19

12  417 3 51 3 3

2  148 1  2 13 2

Solve.

Simplify completely. Assume all variables represent positive real numbers.

21) 13  112

8 17  3

4 4 26) 2h3 1h  16 2h13 4 2 14h3 2 h

Chapter 10 Radicals and Rational Exponents

37) 15h  4  3

{1}

38) z  11  4z  5

{2}

39) 13k  1  12k  1  1

{1, 5}

V , V represents the volume of A ph a cylinder, h represents the height of cylinder, and r represents the radius.

40) In the formula r 

a)

A cylindrical container has a volume of 72p in3. It is 8 in. high. What is the radius of the container? 3 in.

b)

Solve the formula for V.

V  pr2h

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2 1) Combine like terms. 4x  3y  9  x  y  1 3

16) Solve 6g  1 11. Write the answer in interval notation. (q, 2]´ c 53 , qb

2) Write in scientific notation. 8,723,000 8.723 106

17) Solve using Gaussian elimination.

3) Solve 3(2c  1)  7  9c  5(c  2). e  f 3 4

4) Graph 3x  2y  12. 5) Write the equation of the line containing the points (5, 3) and (1, 2). Write the equation in slope-intercept form. 6) Solve by substitution. 2x  7y  12 x  4y  6 (6, 0) 7) Multiply (5p2  2)(3p2  4p  1)

15p4  20p3  11p2  8p  2

8n3  1 8) Divide. 4n2  2n  1 2n  1 Factor completely.

9) 4w2  5w  6 10) 8  18t2

(4w  3) (w  2) 2(2  3t) (2  3t)

3

x  3y  z  3 2x  y  5z  1 x  2y  3z  0

18) Simplify. Assume all variables represent nonnegative real numbers. a)

1500

10 15

b)

3 1 56

3 21 7

c)

2p10q7

p5q3 1q

d)

4 2 32a15

4 2a3 2 2a3

9 1 3

b)

843 16

19) Evaluate. a)

8112

c)

(27) 13

20) Multiply and simplify 213(5  13). 1013  6 21) Rationalize the denominator. Assume the variable represents positive real numbers.

11) Solve 3(k2  20)  4k  2k2  11k  6. {6, 9} 12) Write an equation and solve. The width of a rectangle is 5 in. less than its length. The area is 84 in2. Find the dimensions of the rectangle. legnth  12 in., width  7 in. Perform the operations and simplify.

13)

5a2  3 3a  2  a4 a2  4a

10m2 6n2 14)  9n 35m5 15) Solve

2a2  2a  3 a(a  4)

(4, 1, 2)

a) c)

20 A 50 x 3

110 5

b)

3 x1 y

2y

2

d)

y

6

3 31 4

3

12 1a  2 1  1a

a  2  1a 1a

22) Solve. a)

12b  1  7  6



b)

13z  10  2  1z  4

{3}

4n 21m3

3 4   1. {8, 4} r3 r  8r  15 2

Chapter 10 Cumulative Review

703

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