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CHAPTER
10
Radicals and Rational Exponents
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
10.6 Combining Multiplication, Addition, and Subtraction of Radicals •••
in a residential neighborhood
10.7 Dividing Radicals •••
where the speed limit is 25 mph.
10.8 Solving Radical Equations •••
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
627
1 is the square root symbol or the radical sign. The number under the radical sign is the radicand. Radical sign S 125 c Radicand
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 19.
Solution Recall that to find 19 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, 19 is not a real number.
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Chapter 10 Radicals and Rational Exponents
You Try 3 Find 136.
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: 125 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
629
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) 181 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
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b) 3
NetTutor Self-Test
e-Professors
Videos
*Additional answers can be found in the Answers to Exercises appendix.
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) 14
20) 1144
7
22) 181
1 13 not real
24) 136 26) 1100
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) 116
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 a1n 2.
3.
4.
5.
Define and Evaluate Expressions of the Form amn Define and Evaluate Expressions of the Form amn 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) 912 b) 2713 c) 3215 answer: a) 3 b) 3 c) 2
1. Define and Evaluate Expressions of the Form a1n 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) 813 b) 4912 c) 8114
Solution a) The denominator of the fractional exponent is the index of the radical. Therefore, 3 813 1 8 2.
b) The denominator in the exponent of 4912 is 2, so the index on the radical is 2, meaning square root. 491 2 249 7. 4 c) 8114 1 81 3
You Try 1 Write in radical form and evaluate. a) 1614
b) 1211 2
2. Define and Evaluate Expressions of the Form a mn 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, amn (a1n ) 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 amn in this way: amn (am ) 1n 1am. n
Example 2 In-Class Example 2 Write in radical form and evaluate. a) 823 b) 1632 answer: a) 4 b) 64
Write in radical form and evaluate. a) 2532 b) 6423
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. 2532 (251 2 ) 3 ( 125) 3 53 125
Use the definition to rewrite the exponent. Rewrite as a radical. 225 5
b) To evaluate 6423, first evaluate 6423, then take the negative of that result. 6423 (6423 ) (6413 ) 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) 3225
b) 10032
3. Define and Evaluate Expressions of the Form a mn Recall the definition of a negative exponent from Section 2.4. If n is any integer and a 0, then
1 n 1 an 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 24 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 mn 1 amn a b mn . 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) 93 2 b) 8134 8 43 c) a b 27 1 1 answer: a) b) 27 3 81 c) 16
Rewrite with a positive exponent and evaluate. a) 3612
c) a
b) 3225
125 23 b 64
Solution a) To write 3612 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 25 b 32 1 2 a5 b A 32 1 2 a b 2 1 4
3225 a
b)
c)
a
125 23 64 23 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) 1441 2
c) a
b) 1634
8 23 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) (325 ) 6 b) 2723 2713 4133 c) 7 3 4 answer: a) 3125 b) 3 c) 16
Simplify completely. The answer should contain only positive exponents. 829 a) (615 ) 2 b) 2534 2514 c) 119 8
Solution a) (615 ) 2 625 3 1 b) 2534 2514 254 14 2
Multiply exponents. Add exponents. 3 1 Add a b. 4 4
24
25
2 Reduce . 4 Evaluate.
2512 5 c)
82 9 8
119
89 9
Subtract exponents.
899
Subtract
2
11
2 11 . 9 9
9 Reduce . 9
81 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) 4938 4918
b) (16112 ) 3
c)
725 745
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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) x14 x54 c)
h23 h34
b) a
32
q
h53
answer: a) x32 c)
p1 6
b)
8
b
p43 q12
1 74
h
637
Simplify completely. Assume the variables represent positive real numbers. The answer should contain only positive exponents. x23 6 b) a 14 b y
a) r18 r38
Solution 1 3 a) r18 r38 r 8 8 48 r r12
c)
n56 n13 n16
Add exponents.
x23 6 x3 6 b) a 14 b 1 6 y y4 x4 32 y 5 1 56 13 n n n6 3 c) 16 n16 n 5 2 n6 6 16 n n36 16 n 3 1 n6 16 2 2
26
n
n13 1 13 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) (a3b15 ) 10
b)
t310 t710
c)
s34 s12 s54
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 9 948 912 3 6 4 2s s46 s23
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 (712 ) 2 1 72 2 71 7 3 3 (2 5 ) (53 ) 13 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 ann a1 a n 1 an ann 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 912. 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 912, 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 912 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 813. Using the properties in this chapter, we get 3 813 1 8 2.
1.
To evaluate 813 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.
4912
5.
1213
2.
14412
6.
6413
3.
8114
7.
1632
4.
3215
8.
2743
Answers to You Try Exercises 1) a) 2
b) 11
5) a) a30b2
b)
b) 1000
2) a) 4 1 2/5
t
c) s32
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)
1 725
c) t
Answers to Technology Exercises 1. 7
2. 12
3. 3
4. 2
5. 5
7. 64
Practice Problems
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10.2 Exercises
6. 4
8. 81
NetTutor Self-Test
e-Professors
Videos
*Additional answers can be found in the Answers to Exercises appendix.
Objective 1 12
1) Explain how to write 25
9) 1251/3 in radical form.
2) Explain how to write 113 in radical form. video
Write in radical form and evaluate. 3) 912
3 13
4) 6412
8
13
5) 1000
10
6) 27
3
7) 3215
2
8) 8114
3
12
5
10) 641/6 12
11) a
4 b 121
2 11
4 12) a b 9
13) a
125 13 b 64
5 4
14) a
15) a
36 12 b 169
6 13
16 14 b 81
16) a
100 12 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)
in radical form.
3/2
18) Explain how to write 100
in radical form.
61)
Write in radical form and evaluate. video
19) 843
16
20) 8134
27
21) 12523
25
22) 3235
8
56
23
32
24) 1000
100
25) 2743
81
26) 3632
216
16 34 27) a b 81
8 27
64 23 b 28) a 125
16 25
1000 23 b 29) a 27
100 9
8 43 30) a b 27
16 81
35) 100013 1 14 37) a b 81 39) a video
1 13 b 64
41) 6456 43) 12523 25 32 b 4
45) a 47) a
23
64 b 125
34) 10012 36) 2713
3
1 15 38) a b 32
4
40) a
1 32 1 25
42) 8134
8 125
46) a
25 16
1 13 b 125
44) 6423
48) a
9 32 b 100 34
81 b 16
j
71)
20c23 56
72c
73) (q
45 10
)
79) (81u83v4 ) 34 f 67
x53
2
5
87)
1 27 1 16 1000 27
89) a
6
y12 y13
1
y56
y23
a4b3 25 b 32a2b4
video
6 91) 2493
7
57)
443 532 5
92
1 125
58)
234 3235 32
75
1 16
2
2
3
3
4
4
99) ( 1 12) 101) ( 1 15) 3
32
32x5y2
84) a
88)
a125 4b
25
34
16c8
8b114
b 113
c6
b
t32
4
b u14
t6u
t5 t12 t34
90) a
25a6b16
t154
16c8d3 32 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
2234
1
Objective 5
51) (914 ) 2
56)
5w110
80) (64x6y125 ) 56
86) a
x10w9
97) 232
16
f 27g59 3
b w3 2
8
4103
24
25
n6 7 43 52 78) (r ) r103 27u2v3
13
85) a
613
10w
1
81) (32r13s49 ) 35 8r15s415 82) (125a9b14 ) 23 b 27g53
49
48w310
76) (n27 ) 3
x23 z215
83) a
25
h
z
34
74) (t38 ) 16 t6
1
77) (z15 ) 23 video
72)
q
75) (x29 ) 3
1 10 1 3
8 27
h712 68) (3x13 )(8x49 ) 24x19 x16 1 70) 56 x23 x
8
49) 223 273
55)
4
465 435
66) h
18c32
95) ( 15)
54) 643 653
1 75
j310
5
93) 21000
84 5
425
16
a
a49
Simplify completely. The answer should contain only positive exponents.
53) 875 835
1000
64) z16 z56
19
6
52) 1723 2 3
1052 1032 104
k52 1
72v118
a59
Objective 4
50) 534 554
62)
67) (9v58 )(8v34 )
Rewrite with a positive exponent and evaluate. 33) 4912
713
310
65) j
69)
1 32 1 23 b a b 32) a 100 100 1 7 1 10
60)
729
35
Decide whether each statement is true or false. Explain your answer. 9
749 719
63) k74 k34
Objective 3
31) 81
6
6
6
52
Simplify completely. Assume the variables represent positive real numbers. The answer should contain only positive exponents.
23) 64
12
61 12
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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644
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 ) 12 r6 2 r62 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 am2
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 z22 z1 z b) 249t2 149 2t2 7 t22 7t 14 c) 218b 118 2b14 19 12 b142 3 12 b7 3b7 12 32 132 d) A n20 2n20 116 12 n202 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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648
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 x62 1x x3 1x
6 is the largest number less than 7 that is divisible by 2. Product rule Use a fractional exponent to simplify. 623
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 c102 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 p162 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 824 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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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
Boost your grade at mathzone.com!
NetTutor Self-Test
e-Professors
653
Videos
*Additional answers can be found in the Answers to Exercises appendix.
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 27 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 86 3 3 1 8 1 6 3 2 16
Product rule 3 1 82
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 amn. 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 244 84 2t 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 x23 x12 x46 x36 4 3 x6 6 7/6 x 6 7 2 x 6 x1 x
Change radicals to fractional exponents. Get a common denominator to add exponents. Add exponents. Rewrite in radical form. Simplify.
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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
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1 3
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*Additional answers can be found in the Answers to Exercises appendix.
1) In your own words, explain the Product rule for radicals. Answers will vary.
23)
2) In your own words, explain the Quotient rule for radicals. Answers will vary.
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
Section 10.5 Adding and Subtracting Radicals Objectives 1. Add and Subtract Radical Expressions 2. Simplify Radical Expressions Containing Integers, Then Add or Subtract Them 3. Simplify Radical Expression Containing Variables, Then Add or Subtract Them
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
1. Add and Subtract Radical Expressions In order to add or subtract radicals, they must be like radicals.
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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Section 10.5 Adding and Subtracting Radicals
Example 1 In-Class Example 1 Add. a) 3x 11x b) 9 15 215 answer: a) 14x b) 1115
Add. a) 4x 6x
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
Commutative property Add; distributive property Add.
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.
Steps for Adding and Subtracting Radicals
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)
Combine like radicals.
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 82 3 3 Multiply. 14 1 5 1 5 3 Add like radicals. 13 1 5
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Section 10.5 Adding and Subtracting Radicals
665
You Try 3 Perform the operations and simplify. a) 713 112
b) 2163 11128 2121
3 3 c) 1 54 51 16
3. Simplify Radical Expressions Containing Variables, Then Add or Subtract Them We follow the same steps to add and subtract radicals containing variables.
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
632
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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666
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
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10.5 Exercises
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*Additional answers can be found in the Answers to Exercises appendix.
Objectives 1–3
39) 4d2 1d 242d5
1) How do you know if two radicals are like radicals?
40) 16k4 1k 132k9
They have the same index and the same radicand.
2) What are the steps for adding or subtracting radicals? 1) Write each radical expression in simplest form. 2) Combine like radicals.
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
667
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 23 (10 16) 16 (5 2) Multiply. Multiply. 6 10 16 16 10 Combine 4 916 like terms.
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Section 10.6 Combining Multiplication, Addition, and Subtraction of Radicals
669
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 45 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 xy
Substitute 1x for a and 1y for b. Square each term.
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Section 10.6 Combining Multiplication, Addition, and Subtraction of Radicals
671
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
Boost your grade at mathzone.com!
NetTutor Self-Test
e-Professors
Videos
*Additional answers can be found in the Answers to Exercises appendix.
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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video
Chapter 10 Radicals and Rational Exponents
41) (5y2 4) 2
42) (3k 2) 2 9k2 12k 4
65) ( 1c 1d)( 1c 1d)
cd
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 49
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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Section 10.7 Dividing Radicals
673
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 22 12 4
Simplify; quotient rule Product rule 14 2 Rationalize the denominator. 12 12 2 Multiply.
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Section 10.7 Dividing Radicals
675
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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Section 10.7 Dividing Radicals
677
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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679
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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Section 10.7 Dividing Radicals
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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682
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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Section 10.7 Dividing Radicals
683
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
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Videos
*Additional answers can be found in the Answers to Exercises appendix.
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 uv
b 25 1b 5 1b 5
92)
d9 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?
The radical is eliminated.
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
fg
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
685
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 n2
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 (x12 ) 2 x 3 3 (1 x) 3 x since ( 1 x) 3 (x13 ) 3 x 4 4 4 ( 1x) x since ( 1x) 4 (x14 ) 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 c9
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 n2
Step 2: Cube both sides to eliminate the cube root. 3 (1 n) 3 23 n8
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 n2 ? 3 1 82
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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Section 10.8 Solving Radical Equations
687
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 y34
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
x40 x4
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 ? 20 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 a3
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 1a8 9a
Distribute. Subtract 7a. Add 8.
Step 6: Verify that a 9 satisfies the original equation. The solution set is {9}.
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Section 10.8 Solving Radical Equations
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 9x7
Square both sides.
Steps 3 and 4 don’t apply. Step 5: Solve the equation. 9x7 16 x
Add 7 to each side.
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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Section 10.8 Solving Radical Equations
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
w20 w2
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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Section 10.8 Solving Radical Equations
10.8 Exercises
Practice Problems
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e-Professors
693
Videos
*Additional answers can be found in the Answers to Exercises appendix.
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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694
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.
236 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 a1n 1 a.
3 813 1 8 2.
p. •••
If a is a nonnegative number and m and n are integers such that n is positive, then n amn (a1n ) m ( 1 a) m
4 1634 ( 1 16) 3 23 8
p. •••
If a is positive number and m and n are integers such that n is positive, then 1 mn 1 amn a b mn . a a
2532 a
1 32 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 am2.
72 272 B 25 225 236 22 5 622 5
p. ••• Quotient rule Product rule; 125 5 136 6
2k18 k182 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 x82 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 824 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 85 3 3 18 1 5 3 21 5
Product rule 3 1 82
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 amn. 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 a124 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 2 155 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
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Definition/Procedure
Example
Reference
Adding and Subtracting Radicals Like radicals have the same index and the same radicand. In order to add or subtract radicals, they must be like radicals. Steps for Adding and Subtracting Radicals 1) Write each radical expression in simplest form. 2) Combine like radicals.
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) 29 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 t1 Check t 5 and t 1 in the original equation. t 5: t 2 22t 1 5 2 12(5) 1 5 2 19 523 True
t 1: t 2 12t 1 1 2 12(1) 1 121 13 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
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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) 116
3) 181
9
4)
3 5) 1 64
4
5 6) 1 32
3
7) 11 9) 164
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 912. 1 2 1 12 a b
9
6
16) 3215
2
27 13 b 17) a 125
3 5
16 12 18) a b 49
4 7
19) 1614
2
20) 100013
10
23
21) 125 23) a
25
23
64 b 27
1 9 1 27
12
25) 81
27) 8134 29) a
16 9
23
27 b 1000
100 9
35
22) 32 24) a
23
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) 100023
65) 211x5 211x8 11x6 1x 66) 25a2b 215a6b4 5a4b2 13b 67)
64 125
2200k21
10k8
22k
5
3 69) 1 16
2 12
31) 367 387
9
32) (1694 ) 18
13
4 71) 1 48
33) (815 ) 10
64
34)
82
1 32
1
36) (2k56 )(3k12 )
72 753 713
37) (64a4b12 ) 56 39) a 700
8
32a103b10 38) a
81c5d 9 14 2c b 16c1d2 3d74
113
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 13 26) a b 27 32
49) 11000
163 17
Evaluate. 15) 3612
217
50)
13) Explain how to write 823 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 k13 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)
z4 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) 116
3) 127 Evaluate.
5) 161/4 7) 1492
6) 274/3
2
2/3
8 8) a b 125
1 7
1/2
m5/8
11) (2x3/10y2/5 ) 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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Cumulative Review: Chapters 1–10 *Additional answers can be found in the Answers to Exercises appendix.
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)
843 16
19) Evaluate. a)
8112
c)
(27) 13
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 a4 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 1a
22) Solve. a)
12b 1 7 6
b)
13z 10 2 1z 4
{3}
4n 21m3
3 4 1. {8, 4} r3 r 8r 15 2
Chapter 10 Cumulative Review
703
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