Rationals, Irrationals, and Radicals

UNIT 9 Rationals, Irrationals, and Radicals People of ancient times used a rope with knots for measuring right triangles. 332 UNIT 9 RATIONALS, I...
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UNIT 9

Rationals, Irrationals, and Radicals

People of ancient times used a rope with knots for measuring right triangles.

332

UNIT 9

RATIONALS, IRRATIONALS, AND RADICALS

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Are rational numbers very levelheaded? Are irrational numbers hard to reason with? Not really, but rational and irrational numbers have things in common and things that make them different.

Big Ideas ►

A number is any entity that obeys the laws of arithmetic; all numbers obey the laws of arithmetic. The laws of arithmetic can be used to simplify algebraic expressions.



If you can create a mathematical model for a situation, you can use the model to solve other problems that you might not be able to solve otherwise. Algebraic equations can capture key relationships among quantities in the world.

Unit Topics ►

Rational Numbers



Terminating and Repeating Decimals



Square Roots



Irrational Numbers



Estimating Square Roots



Radicals with Variables



Using Square Roots to Solve Equations



The Pythagorean Theorem



Higher Roots

RATIONALS, IRRATIONALS, AND RADICALS

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Rational Numbers The rational numbers are a subset of the real numbers. Examples of rational numbers are 3 , and 23.7__ 1 , 5, −3, 0.25, 10 __ __ 8. 4 2 DEFINITION

( )

a, A rational number is any number that can be expressed as a ratio __ b where a and b are integers and b ≠ 0.

The letter  represents the set of rational numbers. Integers are rational numbers because you can write them as a fraction (a ratio) with a −6 denominator of 1. For example, −6 = ___ 1 . A proper fraction is a fraction where the numerator is less than the denominator. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. Improper fractions have values greater than or equal to 1 and can be written as mixed numbers. A mixed number is a number consisting of both a whole number and a fraction or the opposite of such a number. Proper Fractions

7 __ 3 1 __ 1 __ −__ 8, 3, 2, 4

Improper Fractions

26 __ 16 ___ 4 ___ 12 −___ 5 , 4, 8 , 5

NOTATION The letter  denotes the set of rational numbers.

Mixed Numbers

7 2 , 1 __ 1 __ −3 __ 3 4, 5 8

Writing Rational Numbers Example 1 A.

25 Write −___ 8 as a mixed number.

Solution Write the improper fraction as a mixed number by dividing the numerator by the denominator. Because a mixed number has a whole 25 number part, and whole numbers are not negative, consider −___ 8 as the 25 opposite of ___ 8. (continued)

RATIONAL NUMBERS

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( )

25 1 __ − ___ 8 = − 38

( )

1 = −3__ 8 B.

Divide. Simplify. ■

Write 4.25 as a percent, a mixed number, and an improper fraction.

REMEMBER

Solution

To write a decimal as a percent, move the decimal point to the right two places and add zeros if necessary.

Write 4.25 as a mixed number.

Write 4.25 as a percent. 4.25 = 425%

25 4.25 = 4 ____ 100 1 = 4 __ 4

Write 0.25 as a fraction. Simplify.

Write 4.25 as an improper fraction. 1 4.25 = 4 __ 4

Write the decimal as a mixed number.

4·4+1 = ________ 4

Multiply the whole number by the denominator and add it to the numerator.

16 + 1 = ______ 4

Multiply.

17 = ___ 4

Add. ■

Comparing Rational Numbers Two rational numbers are either equal or not equal to each other. If they are not equal, then one of the numbers is greater than the other number.

COMPARISON PROPERTY OF RATIONAL NUMBERS For positive integers a and c and nonzero integers b and d, a > __ c if and only if ad > bc. __ b

d

a < __ c if and only if ad < bc. __ b

d

c if and only if ad = bc. a = __ __ b

d

Example 2 Write to make a true statement. A.

3 __

7 __

4

9

7 3 __ Compare the rational numbers __ 4 and 9 . Use the comparison c 7 a 3 and __ = __ . property of rational numbers, with __ = __ b 4 d 9

Solution

ad = 3 · 9 = 27 and bc = 4 · 7 = 28 7 3 __ Since 27 < 28, __ 4 < 9. ■ 336

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RATIONALS, IRRATIONALS, AND RADICALS

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B.

23 −___ 9

18 −___ 7

Solution Write each fraction as a mixed number.

( )

( )

23 18 _5_ _5_ ___ _4_ _4_ − ___ 9 = − 2 9 = −2 9 and − 7 = − 2 7 = −2 7

( )

( )

5 Compare _9_ and _47_. Multiply so you can use the comparison property of rational numbers, ad = 5 · 7 = 35 and bc = 9 · 4 = 36. Since 35 < 36, _5_ _4_ _5_ _4_ 9 < 7 . That means −2 9 is closer to 0 on the number line than −2 7 and 23 5 18 ___ −2 _9_ > −2 _74_. Therefore, −___ 9 >−7. ■

THINK ABOUT IT On a number line, a number to the right of another number is the greater of the two numbers.

Finding a Rational Number Between Two Rational Numbers The density property of rational numbers states that there are infinitely many numbers between any two rational numbers. Example 3

3 3 Find a rational number between _4_ and _5_.

Solution One solution is to find the number halfway between the two numbers. Find the average of the two numbers by finding their sum and dividing by 2. Step 1 Add the numbers.

Step 2 Divide the sum by 2.

_3_

27 ___

_3_ _3_ _5_ _3_ _4_ 4+5=4·5+5·4

27 _2_ ___ 20 ÷ 2 = 20 ÷ 1

TIP

= 20 + 20

= 20 · 2

Check the answer to Example 3 by writing each fraction as a decimal.

27 = ___ 20

27 = ___ 40

3 = 0.6 3 = 0.75, ___ 27 = 0.675, and __ __

15 ___

12 ___

27 _1_ ___

4

40

5

27 _3_ _3_ The rational number ___ 40 is between 4 and 5 . ■

Proving that the Rational Numbers Are Closed Under a Given Operation Recall that a set of numbers is closed under an operation if the result of the operation with two numbers in the set is also a member of that set. The rational numbers are closed under addition, subtraction, multiplication, and division. Example 4 Prove that the rational numbers are closed under multiplication. Solution For all integers a and c and nonzero integers b and d, c ac ac _a_ · __ = ___ . The number ___ is a rational number because the set of b d bd bd integers is closed under multiplication. The denominator cannot be zero because neither b nor d is zero. ■

RATIONAL NUMBERS

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Problem Set Write each number as a mixed number, proper or improper fraction, decimal, and percent, if possible. 1.

7 ___

2.

2.8

20

3.

0.3125

5.

1 11__ 8

4.

15 ___

6.

13 ___

2

5

Write to make a true statement. 5 −__ 6

7. 8.

8 ___

11 ___

13

16

1 12 __ 3

9.

6 −__ 7

37 ___ 3

36 ___

29 ___

15

8

11.

5 7 __ 8

31 ___

12.

−0.71

10.

13. 14.

4

25 −___ 19

4 −1___ 17

13 ___

169 ____

14

182

7 −__ 9

Arrange each list of numbers in increasing order. 7 ___ 8 ___ 11 __

15. 16.

8 , 10 , 13

18.

7 5 3 __ __ −___ 13 , − 5 , − 8

21.

9 ___ 13 2 __ −3__ 3, −2, − 3

19.

27 64 ___ 2.58, ___ 25 , 10

22.

20.

130 ____ 140 ___ 45 ___ 52 −____ 63 , − 46 , − 6 , − 9

7 5 __ 3 __ 1 __ __ 3, 5, 7, 9

17.

69 64 ___ ___

9 , 10 , 6.4

99 ___ 79 53 ____ ___ 68 , 104 , 85

Find a rational number between the two given numbers. 6, 9

25.

47 2 ___ 9 __ 3, 8

17 14 ___ −___ 31 , −33

26.

39 4 ___ −3__ 5 , − 11

5 __ 5 __

23. 24.

27.

2 1, __ 3

Solve. Prove that the rational numbers are closed under:

28.

Challenge 7 __

2 __

8

3

A. addition

A. Find a rational number between − and − .

B. subtraction

B. Determine if your answer is less than −0.7. If

C. division

Challenge

*29.

*30.

A. Find a rational number between

10 ___ 5 and 13.

3 __

7 not, find a rational number between −__ 8 and 2 −__ 3 that is less than −0.7.

B. Determine if your answer is greater than 0.7.

3 If not, find a rational number between __ 5 10 and ___ that is greater than 0.7. 13

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Terminating and Repeating Decimals There are different types of decimals. DEFINITIONS

TIP

Terminating decimals are decimals that have a finite number of nonzero digits. Nonterminating decimals are decimals that do not terminate, or end.

The word terminate means stop. Terminating decimals stop; nonterminating decimals do not stop.

There are two types of nonterminating decimals: repeating and nonrepeating decimals. Repeating decimals have a repeating pattern of digits, while nonrepeating decimals do not. Place a bar over the block of digits that repeat in a repeating decimal. Terminating Decimals _

0.25, 6, −1.5836

NOTATION

Nonterminating Decimals Repeating

0.4___ = 0.44444. . . 3.256_= 3.256256. . . 20.763 = 20.76333. . .

A bar placed over digits in a decimal shows the digits repeat.

Nonrepeating

0.356987412569112. . . 3.1415926535. . .

Converting Fractions to Decimals Example 1 Express each fraction as a decimal. Determine if the decimal repeats or terminates. A.

7 ___ 25

Solution Divide 7 by 25: 7 ÷ 25 = 0.28. The decimal is a terminating decimal. ■ B.

TIP On a calculator, the last digit displayed may be rounded up, even when the digits continue to repeat.

7 __ 9

_

Solution Divide 7 by 9: 7 ÷ 9 = 0.777777. . . = 0.7. The decimal is a repeating decimal. ■ C.

REMEMBER

5 ___ 12

_

Solution Divide 5 by 12: 5 ÷ 12 = 0.41666666. . . = 0.416. The decimal is a repeating decimal. ■

Place the bar over the repeating part of the decimal only.

TERMINATING AND REPEATING DECIMALS

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Expressing Terminating and Repeating Decimals as Fractions A number is rational if and only if you can write it as a terminating or repeating decimal. Both terminating and repeating decimals can be written as the quotient of two integers. Example 2 A.

Write each decimal as a quotient of two integers.

4.25

Solution Since the last digit in the decimal part is in the hundredths place, write 0.25 as a fraction with a demonimator of 100. 17 25 1 1 ___ __ __ 4.25 = 4 + 0.25 = 4 + ____ 100 = 4 + 4 = 4 4 = 4 ■ __

B.

4.25

Solution Write the number as an equation. Multiply each side of the equation by the power of 10 that has as many zeros as there are digits in the repeating block. __

x = 4.25

__

100x = 425.25 Subtract the first equation from the second. This will eliminate the repeating part. Then isolate the variable and simplify the fraction if possible. __

100x = 425.25 −

x=

__

4.25

99x = 421

TIP

99x ____

When multiplying both sides of the equation by a power of 10, it helps to write the repeating decimal without the bar and display the digits in the repeating block a couple of times.

421 ____ 99 = 99 421 x = ____ 99

__

Check 421 ÷ 99 = 4.25  ■ ___

C.

2.02342

Solution

Multiply each side of the equation by 1000. ___

x = 2.02342

___

1000x = 2023.42342 Subtract the first equation from the second. ___

1000x = 2023.42342 −

x=

___

2.02342

999x = 2021.4 Multiply each side by 10 to eliminate the decimal. Then isolate the variable and simplify. 9990x = 20,214 20,214 9990x ______ ______ 9990 = 9990 1123 x = _____ 555 ■ 340

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RATIONALS, IRRATIONALS, AND RADICALS

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Finding a Rational Number between a Fraction and a Repeating Decimal __ 8 Find a rational number between ___ 81. and 0. 11

Example 3 Solution

Step 1 Write the repeating decimal as a fraction. __

100x = 81.81

__



x = 0.81 99x = 81 9 81 ___ x = ___ 99 = 11

Step 2 Find the average of the two fractions. 9 17 8 ___ ___ Add the numbers. ___ 11 + 11 = 11 17 17 __ 17 1 ___ ___ Divide the sum by 2. ___ 11 ÷ 2 = 11 · 2 = 22 __ 17 8 ___ 81. The number ___ is between and 0. 22 11

17 8 ___ Step 3 Check. Write ___ 11 and 22 as decimals. __ __ 17 ___ 27 and 27 = 0.7 = 0.77 11 22

8 ___

__

__

__

0.727 < 0.7727 < 0.81 ■

Application: Proof with Repeating Decimals _

Prove that 0.9 = 1.

Example 4

_

Solution Write 0.9 as a fraction. _

Let x = 0.9.

_

10x = 9.9 _



x = 0.9 9x = 9 9 x = __ 9=1 _

_

Since x = 0.9 from the first assumption, and x = 1 from the algebra, 0.9 = 1 by the substitution property of equality. ■

TERMINATING AND REPEATING DECIMALS

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Problem Set Express each fraction as a decimal. Determine if the decimal repeats or terminates. 1.

3 __

2.

2 ___

3.

1 __

4 13 9

4.

9 __

5.

4 ___

6.

7 __

5 15

7.

23 ___

8.

10 ___

6 27

2

Write each decimal as a quotient of two integers. 7.2

12.

10.

5.13

11.

0.125

9.

270.35

15.

2.54

13.

0.074

16.

3.2

14.

1.72

17.

4.07

26.

_ 11 −3.5 and −___ 3

27.

__ 7 −___ 72 and −0.22 24

30.

1999.9 = 2000

_

___

_

__

_

Find a rational number between the two given numbers. __ _ 5 __ 3 and 0.45 18. ___ 22. 0.136 and 6 10 _

__

19.

3 and 1.8 __

23.

11 and −1.318 −___ 6

20.

__ 10 0.681 and ___ 11

24.

_ 1 0.083 and ___ 13

21.

_ 1 0.16 and __ 3

25.

32 and 2.4 ___

29.

19.9 = 20

2

_

15

Prove each statement. _

_

2.9 = 3

28.

_

Solve and show your work. Challenge

*31.

A. Write

13 ___

1 __ 50 and 4 as decimals.

B. Find a repeating decimal between these two

numbers. C. Write your answer for Part B as a fraction.

342

UNIT 9

*32.

Challenge A. Write

7 ___

8 ___ 11 and 11 as decimals.

B. Find a repeating decimal between these two

numbers. C. Write your answer for Part B as a fraction.

RATIONALS, IRRATIONALS, AND RADICALS

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Square Roots The result of multiplying a number by itself is the square of the number. DEFINITION A square root is the factor of a number that when multiplied by itself results in the number.

Every positive real number has two square roots, one positive and one negative. If x2 = a, then x · x = a and (−x) · (−x) = a. For example, 9 is the square of 3 because 3 · 3 = 9. Notice also that (−3) · (−3) = 9. Since both 32 and (−3)2 equal 9, both 3 and ___ −3 are square roots of 9. is used, only the positive square root is being When the radical sign √ __ asked for. So, √9 = 3. The positive square root is also called the principal, or nonnegative, square root.

NOTATION

___

The radical sign √ indicates the principal square root.

Finding Square Roots Example 1 A.

THINK ABOUT IT___

Evaluate.

The square root of 100.

Solution The square roots of 100 are −10 and 10 because (−10)2 = 100 and 102 = 100. ■ ___

B.

√ 49

√ can also be The radical sign 2 ___ written as √ to indicate the second, or square, root. Most books omit the 2.

Solution The square roots of 49 are 7 and___−7, but since the radical sign is used, give only the principal square root: √49 = 7. ■ __

C.

√__94

__



4 __ 2 Solution Use the principal square root. Since 3 · 3 = 9 , __ 9 = 3. ■ 2 __ 2 __

____

D.

4 __

−√1.44

Solution Find the opposite of the principal square____ root. The principal square root is 1.2. The opposite of 1.2 is −1.2. So, −√1.44 = −1.2. ■

SQUARE ROOTS

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Like Radicals

__

In the square root expression 3√7 , 3 is the coefficient and 7 is the radicand. When two or more square root expressions have the same radicand, the expressions are like radicals and can be combined by adding or subtracting the coefficients.

THINK ABOUT IT

Like Terms

Like Radicals

__ __ __ 3√___ 7 + 5√___ 7 = 8√7___

3x + 5x = 8x 3y − y = 2y 3√14 − √14 = 2√14

Combining like radicals is similar to combining like terms.

Example 2 Simplify. __

__

__

4√2 + 9√2 + 9√3

A.

Solution __

__

__

__

Identify the like radicals: 4√2 and 9√2 . Add their coefficients. __

4√2 + 9√2 + 9√3 __

__

= 13√2 + 9√3 __

__

The sum is 13√2 + 9√3 . These two expressions cannot be combined because their radicands are different. ■ __

__

−9√5 − 2√5

B.

Solution

The radicals are like radicals. Subtract the coefficients.

__

__

−9√5 − 2√5 __

= −11√5



Computing Products of Square Roots The product of two__square __ root ___expressions is the square root of the product of the radicands: √a · √b = √ab . __

__

Example 3 Multiply √5 · √2 . Solution

__

__

____

___

Multiply the radicands: √5 · √2 = √5 · 2 = √10 . ■

Application: Verifying and Justifying Facts About Square Roots Example 4

__

__

_____

Verify that √a + √b ≠ √a + b .

A.

Solution ___

___

Substitute nonzero numbers for a and b. Let a = 16 and b = 25. _______

√ 16 + √ 25

√16 + 25

4+5

√ 41

9

___

__

≈ 6.4 __

_____

9 ≠ 6.4, so √a + √b ≠ √a + b ■

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B.

Explain why a negative number cannot have a real square root.

Solution A square root of a number is a number that, when multiplied by itself, equals the original number. Any number times itself cannot be a negative number because the product of two positive numbers is positive and the product of two negative numbers is positive. ■

Problem Set Evaluate.

___

___

1.

√ 81

√ 36____

5.

√169

4.

−√0.04

7.

1600

8.

0.25

3.

__

2.

−√9

____

1 ___

121 ____

Find the square roots of each number. 6.

144 ____ 49

Simplify. __

√ 7 + 7√ 7

10.

9√2 − 3√2

__

__

__

__

___

√3 · √5

12.

√ 2 · √ 1.1

15.

38 √91 __+ √___ 91 __

___

71 ___

−4√2 − 4√2

2√7 + 3√6 + 4√7 + 2√6

21.

√ 5 · √ 3 · √ 6 · √ 10

√__7 · √__6 + √__14 · ___ √3

22.

8√82 + 2√82

23.

( 3√2 + 2√2 − 4√2 ) · √2

24.

√ 1.5 · √ 150

__

___

___

5 __

2 __

1 ___

10 ___

√ 5 · √ 2 + √ 3 · √ 10 ___

___

____

__

___

___

−3√26 + 3√13 + 10√26 − 3√13 ____

__

__

__

__

__

__

17.

19.

___

__

___

16.

___

15√17 + 25√34 − 10√17

20.

__

14.

___

18.

__

11.

13.

___

__

9.

___

√ 2 · √ 0.33 + √ 0.06 · √ 11

25.

___

___

__

__

___

____

___

___

__

__

___

1 √ 30 · √ 42 · √___ 35 1 ___

1 ___

Solve. 26.

64 2 A square has an area of ____ 169 in . Find the side length.

*29.

Challenge Verify that the set of irrational numbers is not closed under multiplication.

*30.

Challenge

___

27.

28.

A rectangle ___ has a length of √43 meters and a width of √21 meters. Find the area of the rectangle. A circle has an area of 256π square units. Find the radius.

__

__

A. Prove that √ 8 = 2√ 2 . __

__

B. Find √ 3 · √ 6 and simplify.

SQUARE ROOTS

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Irrational Numbers Real numbers that are not rational numbers are irrational numbers. NOTATION The letter  denotes the set of irrational numbers.

Unlike a rational number, an irrational number is a real number that cannot a be written in the form __, for any integers a and b. The set of rational numb bers  and the set of irrational numbers  make up the set of real numbers. ⺢: Real Numbers ⺡: Rational Numbers ⺪: Integers ⺧: Whole Numbers

⺙: Irrational Numbers

⺞: Natural Numbers

Determining if a Number Is Rational or Irrational

REMEMBER π = 3.14159. . .

Any decimal that is nonterminating and nonrepeating is an irrational number. The number π is an example of an irrational number. There is no way to convert the decimal into a fraction of integers because there is no repeating block of digits. A perfect square is a rational number whose square root is also rational. 4. The square root of any numExamples of perfect squares are 9, 25, and __ 9 __ ___ ber that is not a perfect square is an irrational number, such as √2 and √14 .

THINK ABOUT IT A decimal cannot accurately represent an irrational number, but a decimal can approximate __ its value. √2 is an exact value; __ 1.414 approximates √2 .

Rational Numbers __

___

___

√ 9 = 3, √ 25 = 5, √ 36 = 6

Example 1 ___

A.

√ 90

Solution

____

B.

√ 256

Irrational Numbers

__ 2= √___

1.414213562. . . √ 14 = 3.741657387. . .

Determine if each number is rational or irrational. ___

Because 90 is not a perfect square, √90 is irrational. ■ ____

Solution 256 is a perfect square because 16 · 16 = 256. √256 is rational. ■ 346

UNIT 9

RATIONALS, IRRATIONALS, AND RADICALS

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____

−√121

C.

Solution The ____square root of 121 is 11. The opposite of 11 is −11, which is rational. −√121 is rational. ■ __

√__51

D.

Solution Because no number times itself __ equals 5, there is no number that 1 1 __ can be multiplied by itself to equal 5 . __ 5 is irrational. ■



___

64 √___ 49

E.

___

______ 64 __ 8 = 1. 142857, Solution Both 64 and 49 are perfect squares. Since ___ = 49 7 ___ 64 ___ 49 is rational. ■





Simplifying Radicals

___

__

__

To simplify radicals, use √ab = √a · √b . When a radical expression contains no radicands with factors that are perfect squares other than 1, the expression is in simplified radical form. Example 2

Simplify.

___

√ 40

A.

Solution Choose two factors of 40 so that one of them is a perfect square. ___

_____

√ 40 = √ 4 · 10 __

___

= √4 · √10 ___

= 2√10

Ten is not a perfect square ___ and there are no ___factors of 10 that are perfect squares other than 1, so √40 simplifies to 2√10 . ■ ____

B.

√ 108

Solution Choose two factors of 108 so that one of them is a perfect square. ____

_____

√ 108 = √ 36 · 3 ___

__

= √36 · √3

TIP Make a list of the first 15 perfect squares to refer to while simplifying radicals.

__

____

= 6√ 3

__

√ 108 simplifies to 6√ 3

It is possible to choose a perfect square factor that is not the greatest perfect square factor. Suppose you chose 4 and 27. ____

_____

√ 108 = √ 4 · 27 __

___

= √4 · √27 ___

= 2 · √27 __

__

= 2 · √9 · √3 __

__

= 2 · 3 · √3 = 6√3 ■

IRRATIONAL NUMBERS

347

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Verifying the Closure Properties for Irrational Numbers The irrational numbers are closed under addition and subtraction. Example 3 Determine if the set of irrational numbers is closed under multiplication. Solution The product of two irrational numbers is not always an irrational number. Here is an example. __

__

__

√3 · √3 = √9 = 3

The set of irrational numbers is not closed under multiplication. ■

Application: Proving That a Square Root Is Irrational The square of an even number is always an even number and the square of an odd number is always an odd number. Use these facts in the following example. __

Example 4 Prove that √2 is irrational.

THINK ABOUT IT a is not a simplified fraction, If __ b then common factors can be divided out until it is, resulting in an equivalent fraction.

__

Assume that √2 is rational. Then it would be possible to represent a the value as a simplified fraction __, where a and b are integers and b is not b zero. Because the fraction is simplified, a and b have no common factors other than 1. Solution

__

√2 = __

a __ b

a ( √2 )2 = __

2

(b)

a2 2 = __2 b 2b2 = a2

Square both sides of the equation. __

__

__

√ 2 · √ 2 = √ 4 = 2 and

a2 = __2 b b b

a __ a __ ·

Means-Extremes Product Property

Since a number with a factor of 2 is an even number, 2b2 is an even number. That means a2 is an even number, and since only an even number squared can result in an even number, a must also be even. Since a is an even number, it can be written as a product with a factor of 2. Let a = 2c, where c is an integer. 2b2 = a2

Last line from above

2b2 = (2c)2

Substitution Property of Equality

2b = 4c

2c · 2c = 4c2

2

2

b2 = 2c2

TIP When an assumption leads to a contradiction (two statements with opposite ideas), the assumption is false.

348

Divide each side by 2.

2

UNIT 9

Now b is shown to be an even number because it is equal to a product with a factor of 2. Since b2 is even, b is even. Both a and b have been shown to be even, which means they both have a common factor of 2. However, it was stated in the beginning of the proof that a and b have no common factors other than 1. This is a contradiction. There__ fore, the assumption that √__2 is rational must be incorrect. A number is either rational or irrational, so √2 is irrational. ■

RATIONALS, IRRATIONALS, AND RADICALS

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Problem Set Determine if each number is rational or irrational. Explain. ___

___

1.

√ 75

4.

−√49

2.

√ 100

5.

√ 144

3.

√7

6.

π

15.

√9

16.

√____ 81

____ __

3 __

___

√____ 9 4 __

√ 100

17.

12.

√ 125

13. 14.

11.

8.

√25

__

√ 72 __

10.

−√170

____

Simplify. 9.

____

7.

___

16 ___

____

21.

___

−√288 __

36 ___

22.

1 __ √____ 4

√ 196

23.

√ 169

18.

−√162

24.

√ 27

−√12

19.

√ 300

25.

5√8

√ 45

20.

√ 121

____

____

___

___

____

___

__

____

Solve. 26.

Show that the set of irrational numbers is not closed under division. That is, show that there are two irrational numbers with a quotient that is not an irrational number.

*27.

Challenge The unit square diagrams show that every perfect square can be written as a sum of the form 1 + 3 + 5 + 7 + . . .

1

4=1+3

*28.

Challenge Write an example to illustrate the process of finding the greatest perfect square divisor of x. Choose a value a and test to see if x x is an integer. If __ __ is an integer, a is the a2 a2 greatest perfect square divisor of x. Begin __ with the least value of a that is greater than √x and work backwards.

9=1+3+5

A. Complete the list below:

1=1 4=1+3 9=1+3+5 16 = 25 = 36 = B. Predict the number of addends in the expan-

sion for 144 and 1024.

IRRATIONAL NUMBERS

349

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Estimating Square Roots REMEMBER Irrational numbers are nonterminating, nonrepeating decimals.

Because the square root of a number that is not a perfect square is an irrational number, its decimal value can only be estimated. PROPERTY For nonnegative values of m, n, and p,

__

__

__

if m < n < p, then √ m < √n < √ p .

If a number is between two other numbers, then its square root is between roots of the other numbers. To illustrate, 3 < 4 < 5 and __ __ the square __ √3 < √4 < √5 . 3⬇1.73

1

2

5⬇2.24

3

4

5

4=2

Determining the Location of the Square Root of a Non-Perfect Square Example 1 root lies.

Determine between which two consecutive integers each square

___

A.

√ 52

Solution Think of the perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ... Choose the nearest perfect ___ squares less ___ than and ___greater than 52. They are 49 and 64. Because √49 = 7 and √64 = 8, √52 lies between 7 and 8. ■ ____

B.

√ 125

2 Solution Since ____ 125 lies between the perfect squares 121 (11 ) and 2 144 (12 ), √125 lies between 11 and 12. ■

350

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Estimating Square Roots of Non-Perfect Squares To estimate the square root of a number, first determine between which two consecutive integers the square root lies. Adjust the estimate based on which perfect square the number is closer to and by how much. ___

Example 2

Estimate √23 to the nearest tenth.

___

Solution Twenty-three lies between 16 and 25, so √23 lies between ___ 4 and 5. Since 23 is about three-quarters of the way between 16 and 25, √23 should be about three-quarters of the way between 4 and 5, or at about 4.75. Test values around 4.75. 4.72 = 22.09 and 4.82 = 23.04 ___

So, √23 ≈ 4.8. ■

Using the Babylonian Method to Estimate a Square Root Another method to use when estimating the square root of a number is the Babylonian method.

USING THE BABYLONIAN METHOD TO FIND THE SQUARE ROOT OF x

THINK ABOUT IT

Step 1 Start with any guess r1 of the square root. x to compute a new guess r . Step 2 Find the average of r1 and __ 2 r1

The first guess is r1, the second guess is r2, and the nth guess is rn .

x are as Step 3 Repeat Step 2 using r2. Continue this process until rn and __ rn close as desired. __

Example 3 Use the Babylonian method to estimate √7 to the nearest hundredth. Solution Step 1 Guess 2.5 because 2 · 2 = 4 and 3 · 3 = 9. x 7 r1 + __ 2.5 + ___ r1 ________ 2.5 + 2.8 ___ 2.5 ________ 5.3 _______ Step 2 = = = 2 = 2.65 2 2 2 x 7 r2 + __ 2.65 + ____ r 2.65 + 2.64 ____ 5.29 2.65 __________ __________ ______2 = Step 3 ≈ ≈ 2 ≈ 2.645 ≈ 2.65 2 2 2 The estimates after Step 2 and Step 3 are so close that__ the hundredths place doesn’t change, so this is a good stopping point. So, √7 ≈ 2.65. ■

ESTIMATING SQUARE ROOTS

351

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Application: Geometry Example 4 Find the perimeter of a square whose area is 450 square meters. Round your answer to the nearest tenth of a meter. Solution The side length of a square is the square root of the area. Perim____ eter is four times the length of one side, so the expression 4√450 represents the perimeter of the square. ____

4√450 ≈ 84.9 The perimeter is about 84.9 meters. 2

= 450.500625 ≈ 450  ■ ( 84.9 4 ) ____

Check

Problem Set Determine between which two consecutive integers each square root lies. ___

____

___

1.

√ 19

4.

√ 240

7.

√ 40

2.

√ 28

5.

√ 70

8.

√ 61

3.

√ 12

6.

√ 92

9.

√ 136

___

___

___

___

___

____

Estimate the given square root to the nearest tenth. ___

___

__

10.

√ 17

14.

√5

18.

√ 23

11.

√ 78

15.

√ 150

19.

√3

12.

√ 45

16.

√ 20

20.

√ 60

13.

√ 85

17.

√ 55

___

____

___

__

___

___

___

___

Use the Babylonian method to estimate the given square root to the nearest tenth. ___

____

√ 30

21.

√ 200

____

___

23.

√ 165

Complete____ the missing information in the problem to find √300 using the Babylonian method.

26.

The area of a square is 45 square units. Estimate the side length.

First Guess: 17

27.

Five times the square of a number is 165. Estimate the number.

28.

The quotient of the square of a number and 2 is 0.6. Estimate the number.

29.

The area of a circle is 20π m2. Find the approximate radius of the circle.

22.

24.

√ 99

Solve. When estimating, round to the nearest tenth. 25.

300 17 + ____

+ 300 ________ 17 ________

2

2

________ =

=

589 ____ 17 ____



.3235;

300.1036523 + 300 _________________ + _______ 17.3235 17.3235 = _________________ ____________ = 2 2 _______ 17.3235 ≈ 17.32 ≈ 17.3 _______

2

352

UNIT 9

*30.

Challenge Square a number, divide the result by 10 and multiply that result by 2 to obtain 57.8. What is the number?

RATIONALS, IRRATIONALS, AND RADICALS

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Radicals with Variables Radical expressions may contain variables. The square root of a nonnegative number is squared __ __ that____ __ is the original 3 = √9 = 3. number. For example, √32 = 3 because √32 = √3 ·__ Suppose the base of the power is__a variable: √a2 . Since the radical sign indicates the principal square root, √a2 = a.

Evaluating a Variable Square-Root Expression When evaluating a variable square-root expression, treat the radical sign as a grouping symbol. Example 1 Evaluate. If necessary, round your answer to the nearest hundredth. ______

√ 11 − a , when a = 2 and when a = −3

A.

Solution

______

__

√ 11 − 2 = √ 9 = 3

_________

___

= √14 ≈ 3.74 ■ √11 − (−3) ___ ______

10 − √2d + 2√d + 15 , when d = 15

B.

Solution

_____

_______

10 − √2 · 15 + 2√15 + 15 ___

___

10 − √30 + 2√30

Simplify under the radical signs.

___

10 + √30

Add like radicals.

≈ 15.48

________________

Find the sum. Then round. ■

4√(b + c) + (b − 5) , when b = 7 and c = 6

C.

2

2

Solution

________________

4√(7 + 6)2 + (7 − 5)2 _______

4√132 + 22

Simplify inside the parentheses.

4√169 + 4

Evaluate the powers.

_______ ____

4√173

≈ 52.61

Add.

TIP Rounding should always be the last step in evaluating a squareroot expression.

Find the product. Then round. ■

RADICALS WITH VARIABLES

353

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Simplifying Square Roots of Powers with Variable Bases Example 2 Simplify each expression. Assume all variables are nonnegative. ____

√x6y2

A.

____

______

√x6y2 = √(x3)2y2 = x3y ____

Solution

√ 16a5

B.

____

_____



_______

__

Solution √16a5 = √16a4a = √42(a2)2a = 4a2√a ■ _______

TIP

2√150p18q3

C.

The square root of a power with an exponent that is an even number is a perfect square.

Solution

_______

___________

2√150p18q3 = 2√52 · 6(p9)2q2q ___

= 2 · 5p9q√6q ___

= 10p9q√6q ■

Simplifying Radical Expressions with Quotients Just as the product of two square roots is the square root of the product, a quotient of two square roots is the square root of the quotient.

PROPERTY For nonnegative values a and b, where b ≠ 0, __

__

√b

√__ a = __ a ___ √b

Example 3 Simplify each expression. __

√__9x

THINK ABOUT IT

A.

x is simplified The expression ___ 3 __ because the radicand √ x does not contain a quotient.

Solution simplify.

__



__

Rewrite the expression as a quotient of two radicals. Then

__

x √__ ___

__

√x ___

√ 9 = √9 = 3 x __



___

B.

√ 72 ____ __ √2

Solution Rewrite the expression as the square root of the quotient of the radicands. ___

___

___ 72 = = ___ √ 36 = 6 ■ 2 √2

√ 72 ____ __

354

UNIT 9



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Rationalizing the Denominator A radical expression is not considered simplified if a radical appears in a denominator. To clear a radical from a denominator, you can rationalize the denominator. To rationalize a denominator, multiply the numerator and denominator of the expression by the radical in the denominator of the expression. Example 4 __

THINK ABOUT IT __

Simplify each expression.

√__7x

A.

7 is __ The expression ___ √7 equivalent to 1, so multiplying an expression by it does not change the value of the expression. √

Solution __

__

x √__ ___

√ 7 = √7 x __

__

= B.

√7

__ __

__

7 x ___ √__ √__ ___ ·

√7

=

√x √ 7___ _____ √ 49

___

√ 7x ____

= 7



−4y ____ __

TIP

√8

Solution −4y −4y −4y −2y ____ __ = _____ __ __ = ____ __ = ____ __ 2√2 √4 √2 √2 √8 __ __ __ 2y −2 −2y √ 2 √ ____ ___ ______ = __ · __ = 2 = −√2 y ■ √2 √2

Simplifying the radical in the denominator first will minimize reducing later in the problem.

Application: Analyzing a Bogus Proof Example 5

32 32 __ The statement 1 − __ = 2 − 2 2 is true.

(

) (

)

Find the error in the following proof that 1 ≠ 2. Step 1

2

2

Step 2 Step 3

2

( 1 − __32 ) = ( 2 − __32 ) _______ _______ 3 __ √( 1 − 2 ) = √( 2 − __23 )

Given 2

Take the square root of each side.

3 3 __ 1 − __ 2=2−2

Step 4

1=2χ

3 Add __ 2 to each side and simplify.

Therefore, 1 ≠ 2. 3 3 __ Solution Simplify 1 − __ 2 and 2 − 2. 3 3 __ 1 1 __ __ 1 − __ 2 = −2 and 2 − 2 = 2 This means that in Step 2, when the square root of each side is taken, the square root of a negative number was taken on the left side. This step is not allowed. ■

RADICALS WITH VARIABLES

355

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Problem Set Simplify. Assume all variables are nonnegative.

__

____

√x4y10

1.

8.

√4

15.

√___11x

9.

√ 36 ____ __

16.

2x ____ ___

17.

3x ___ −____ √ 27

___

____

√ 25x

9

2.

3√90a12b15

√ 128 _____ __

11.

√25

18.

x √___ 13

12.

√ 700 _____ __

19.

4x ____ −_____ √ 108

√x y

7 8

√ 72x

5.

√7

___

13.

√121

20.

x √___ 10

14.

√ 80 ____ __

21.

5a ____ ___

√ 64x

13

a ____ ___

________

4√50x y z

14 11 2

7.

___

x ___

____

_____

6.

2 √ ___

____

____ 3

√ 18

10.

____

4.

√4

____

_______

3.

___

x __

√5

√ 84

Evaluate each expression for the given values. Provide answers in simplified radical form and approximate irrational answers to the nearest hundredth.

______

_____

√x + 7

22.

26.

A. x = 10 B. x = −2

_____

√ 2x − 1

A. x = 1 B. x = 3

_______________

___

23.

15 + 2√a + 8 − √3a , when a = 4

27.

3√(ab)2 + (2a + b)3 , when a = 1 and b = 4

24.

2√(x + y) − (x − y)2 , when x = 8 and y = 5

28.

√2x + 3 − 10√x + 1 − 5, when x = −1

25.

20 + √4 − x − 3√x , when x = 2

_______________ _____

______

__

_____

Solve. 1 2 = __ 1−2 Challenge The statement 1 + __ 2 2 is true. Find and explain the error in the following proof.

(

*29.

1 2 = __ 1−2 1 + __ 2 2

) (

2

( ) ( ) _______ _______ 1 __ √( 1 + 2 ) = √( __21 − 2 ) 2

2

1 = __ 1 1 + __ 2 2−2

2

)

*30.

Challenge The statement (1 − 4)2 = (1 + 2)2 is true. Find and explain the error in the following proof. (1 − 4)2 = (1 + 2)2

_______

_______

√(1 − 4)2 = √(1 + 2)2 1−4=1+2 −4 = 2 χ

1 = −2 χ

356

UNIT 9

RATIONALS, IRRATIONALS, AND RADICALS

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Using Square Roots to Solve Equations An equation can have zero, one, two, or even an infinite number of solutions. When a solution of an equation is substituted for the variable, the statement is a true statement.

Using Square Roots to Solve Equations Consider the equation x2 = 9. This equation has two solutions because both 3 and −3, which are the square roots of 9, make the equation true. To solve an equation in which the variable is squared, take the square root of each side __ of the equation. Because √x2 = |x|, __ and both the positive and negative square roots are solutions, write x = ±√9 .

PROPERTY

__

NOTATION The symbol ± is read as “plus or __ minus.” The expression ±√ 9 is an__abbreviated way of writing __ “√ 9 and −√9 .”

For nonnegative values of a, if x2 = a, then x = ±√ a .

Example 1 Solve the equation. If necessary, round your answer to the nearest tenth. A.

n2 = 25

Solution n2 = 25

___

n = ±√25

Take the square root of each side.

n = ±5

Simplify √25 .

___

The solutions are 5 and −5. ■ 5t2 = 31

B.

Solution Isolate the variable and take the square root of each side. 5t2 = 31 31 t2 = ___ 5

Divide each side by 5. ___

√___

31 t = ± ___ 5

Take the square root of each side.

t = ±√6.2

Simplify the radicand.

t ≈ ±2.5

Estimate √6.2 with a calculator.

___

___

___

The solutions are exactly √6.2 and −√6.2 or about 2.5 and −2.5. ■ (continued)

USING SQUARE ROOTS TO SOLVE EQUATIONS

357

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9x2 − 64 = 0

C.

Solution

Use inverse operations to isolate the variable.

9x − 64 = 0 2

9x2 = 64 64 x2 = ___ 9

Add 64 to each side. Divide each side by 9. ___

√___

64 x = ± ___ 9

√ 64 __ x = ±____ √9 8 x = ±__ 3

Take the square root of each side. Write the square root as a quotient. ___

__

Simplify √64 and √9 .

8 8 __ The solutions are __ 3 and −3. ■

Applications: Geometry and Physics Example 2 The area of a circle is 40 square centimeters. Estimate the radius to the nearest tenth of a centimeter. Solve the area formula A = πr 2 for r.

Solution

40 = πr 2

Substitute 40 for A.

40 ___

Divide each side by π.

π = r

2

___



40 ± ___ π = r

Take the square root of each side.

±3.6 ≈ r

Use a calculator to estimate.

Because length cannot be negative, disregard the negative answer. The radius is about 3.6 centimeters. ■ Example 3 Solve −16t2 + 80 = 0 to estimate the number of seconds t it takes for an object dropped from 80 feet above the ground to hit the ground. Solution −16t2 + 80 = 0 −16t2 = −80 t2 = 5

__

t = ±√5 ≈ ±2.236

Subtract 80 from each side. Divide each side by −16. Take the square root of each side.

Since time cannot be negative, disregard the negative answer. The time is about 2.24 seconds. ■

358

UNIT 9

RATIONALS, IRRATIONALS, AND RADICALS

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Problem Set Solve each equation. If necessary, round your answers to the nearest tenth. 1.

x2 = 64

7.

16x2 = 81

13.

6z2 − 183 = 111

Challenge

2.

y2 = 121

8.

25t2 = 144

14.

3x2 = 48

*19.

3.

n2 = 625

9.

4x2 − 49 = 0

15.

4y2 = 14

2(4x + 7) _________ = x2 + 3

4.

t2 = 12

10.

3y2 − 27 = 0

16.

5z2 = 56

*20.

15m2 − 6 = 28.5 _____

5.

x = 96

11.

5m − 140 = 40

17.

4x + 37 = 127

*21.

x2 = 36y

6.

y2 = 196

12.

9x2 + 50 = 219

18.

2(6n2 − 17) = 19

2

2

2

2

7

4

Solve. 22.

Estimate to the nearest hundredth of a second the time it would take a car to travel 50 feet from a stop, at a constant acceleration of 3.2 ft/s2. 1 2 Use d = __ 2 at , where d represents distance, a represents acceleration, and t represents time in seconds.

23.

The area of a circle is 20 square inches. Estimate the radius to the nearest tenth of an inch.

24.

The surface area of a cube is 100 square centimeters. Estimate to the nearest tenth of a centimeter the side length of the cube using the formula S = 6s2, where S represents surface area and s represents side length.

25.

26.

The height of a dropped object can by modeled 1 2 by the formula h(t) = −__ 2 gt + h0 where h represents the height after time t (in seconds), h0 represents the initial height, and g represents the acceleration due to gravity. A. Estimate to the nearest hundredth of a second

the time it takes an object dropped from an initial height of 63 meters to reach the ground on earth (g = 9.8 meters per square second). B. Estimate to the nearest hundredth of a second

the time it takes an object dropped from an initial height of 63 meters to reach the moon (g = 1.62 meters per square second).

The kinetic energy of an object in motion can 1 2 be modeled by the equation E = __ 2mv , where E represents the kinetic energy, m represents the mass, and v represents velocity. What is the velocity (in meters per second) of an object with a mass of 20 kilograms and kinetic energy of 33,640 Joules?

USING SQUARE ROOTS TO SOLVE EQUATIONS

359

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The Pythagorean Theorem Squaring and taking square roots of numbers can help you solve many geometric problems, such as those involving triangles. REMEMBER

DEFINITIONS

A right angle measures 90° and is indicated by a square □.

A triangle with a right angle is a right triangle. The two sides of the triangle that form the right angle are the legs. The side opposite the right angle is the hypotenuse.

hypotenuse

leg

leg

Using the Pythagorean Theorem The Pythagorean theorem states the relationship between the lengths of the sides of a right triangle.

PROPERTY: THE PYTHAGOREAN THEOREM In a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse.

c

a

a2 + b2 = c2

b

360

UNIT 9

RATIONALS, IRRATIONALS, AND RADICALS

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Example 1

Find the value of x.

A.

48 in.

14 in.

x in.

Solution The unknown side is the hypotenuse c. a2 + b2 = c2 14 + 48 = x 2

2

2

196 + 2304 = x

2

THINK ABOUT IT Substitute 14 and 48 for a and b, and x for c.

You would get the same answer by substituting 48 for a and 14 for b.

Simplify the left side.

2500 = x2

_____

±√2500 = x

Take the square root of each side.

±50 = x Disregard the negative answer because lengths must be nonnegative. The value of x is 50. ■ B. 13 m

xm

12 m

Solution The unknown side is a leg. a2 + b2 = c2 x2 + 122 = 132

Substitute x for a, 12 for b, and 13 for c.

x2 + 144 = 169

Evaluate powers.

x2 = 25

___

x = ±√25

TIP The hypotenuse is always the longest side of a right triangle. You know you have made an error if the length of a leg is greater than the length of the hypotenuse.

Subtract 144 from each side. Take the square root of each side.

x = ±5 Disregard the negative answer. The value of x is 5. ■

Using the Converse of the Pythagorean Theorem The Pythagorean theorem tells you how to find the length of a side of a triangle given the triangle is a right triangle. The converse of the Pythagorean theorem tells you how to determine if a triangle is a right triangle given the lengths of all three sides.

THE CONVERSE OF THE PYTHAGOREAN THEOREM If the sum of the squares of the lengths of the shorter sides of a triangle equals the square of the length of the longest side, then the triangle is a right triangle.

(continued)

THE PYTHAGOREAN THEOREM

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Example 2 triangle. A.

Determine if a triangle with the given side lengths is a right

15 ft, 32 ft, and 36 ft

Solution

Substitute 15 for a, 32 for b, 36 for c, and simplify.

a + b = c2 2

2

152 + 322  362 225 + 1024  1296 1249 ≠ 1296 The two sides of the equation are not equal, so the side lengths do not form a right triangle. ■ B.

20 ft, 16 ft, and 12 ft

Solution

Substitute 20 for c, 16 for a, 12 for b, and simplify.

a2 + b2 = c2 162 + 122  202 256 + 144  400 400 = 400  The two sides of the equation are equal. The side lengths form a right triangle. ■

Determining the Relative Measure of the Greatest Angle The greatest angle in a triangle is opposite the longest side c.

PROPERTY

TIP The inequalities c > a and c > b mean that c is the longest side in the triangle.

For a triangle with side lengths of a, b, and c, where c > a and c > b, • if c2 > a2 + b2, then the angle opposite side c has a measure greater than 90°. • if c2 < a2 + b2, then the angle opposite side c has a measure less than 90°.

Example 3 The side lengths of a triangle are 9 m, 12 m, and 14 m. Does the greatest angle in the triangle measure 90°, more than 90°, or less than 90°? Solution

Determine if c2 is less than, greater than, or equal to a2 + b2.

c2

a2 + b2

142

92 + 122

196

81 + 144

196 < 225 Because c2 < a2 + b2, the angle opposite side c, which is the greatest angle, has a measure that is less than 90°. ■

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Application: Boating and Kites Example 4 A boat leaves a dock and travels 3.5 miles due north and 6 miles due west. How far is the boat from the dock? Round your answer to the nearest tenth of a mile. Solution Draw a diagram.

REMEMBER The shortest distance between two points is a straight line.

6 mi

TIP 3.5 mi

x mi

North

West

The unknown distance x is the length of the hypotenuse of a right triangle. Use the Pythagorean theorem.

East

South

a2 + b2 = c2 3.52 + 62 = x2 12.25 + 36 = x2 48.25 = x2

_____

±√48.25 = x ±6.946 ≈ x Disregard the negative answer. The boat is about 6.9 miles from the dock. ■ Example 5 Eduardo is flying a kite on a string that is 75 meters long. The kite is 62 meters above the ground. Estimate the distance between Eduardo and the spot on the ground directly beneath the kite. Solution Draw a diagram.

75 m

62 m

xm

The unknown distance x is the length of a leg of a right triangle. Use the Pythagorean theorem. a2 + b2 = c2 x2 + 622 = 752 x2 + 3844 = 5625 x2 = 1781

_____

x = ±√1781 ≈ ±42.2 Disregard the negative answer. Eduardo is about 42.2 meters from the spot beneath the kite. ■

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Problem Set Find the value of x. If necessary, round your answer to the nearest hundredth. 6.

1.

x

4

267

x 3

2.

15

12

11

7. x

3.

x

7

x

60

4

8.

32

4.

12

x x

55

48

8

5.

4

x

7

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Determine if a triangle with the given side lengths is a right triangle. 10 in., 24 in., and 26 in.

13.

11 in., 60 in., and 61 in.

10.

18 cm, 24 cm, and 30 cm

14.

51 m, 68 m, and 85 m

11.

12 ft, 20 ft, and 24 ft

15.

40 yd, 56 yd, and 70 yd

12.

20 mm, 24 mm, and 36 mm

9.

For a triangle with the given side lengths, determine if the measure of its largest angle is greater than, less than, or equal to 90°. 16.

2 yd, 3 yd, and 4 yd

19.

5 mi, 18 mi, and 20 mi

17.

40 cm, 42 cm, and 58 cm

20.

65 km, 72 km, and 97 km

18.

21 m, 23 m, and 25 m

21.

96 ft, 140 ft, and 165 ft

26.

Mr. Edwards has placed a 20-foot flagpole in his front yard. To help secure the pole, he has strung wires from 2 feet below the top of the flagpole to the ground. The wires are 19 feet long. About how far from the bottom of the flagpole do they reach the ground? Round to the nearest foot.

27.

Taro’s garden is in the shape of a square with an 1 area of 72 __ 4 square yards. She has made a walkway in a straight line from one corner to the opposite corner. About how long is the walkway? Round to the nearest yard.

28.

Jessamyn is in a hot air balloon 63 meters from the ground. A telephone pole is directly below her. An oak tree stands 60 meters from the telephone pole. How far apart are the hot air balloon and the top of the oak tree?

*29.

Challenge A boat is out to sea due east of a marina on the shore. The marina stands exactly halfway between two houses to the north and south, which are 72 miles apart. The boat is 39 miles from each house. How far is the boat from the marina?

*30.

Challenge Ashley rode her bike 6 kilometers due south, turned and rode 4 kilometers due east, and then turned again and rode 8 kilometers due south. To the nearest hundredth of a kilometer, how far is she from her starting point?

For each problem, make a sketch and solve. 22.

Southville is 57 miles due south of Portland. A. Westfield is 76 miles due west of Portland.

How far apart are Southville and Westfield? B. Eastborough is 27 miles due east of South-

ville. About how far is Eastborough from Portland? Round to the nearest mile. 23.

Kenji and Isabel are standing exactly opposite one another on either side of the bank of a 40-meter-wide river. A. Kenji turns and walks 9 meters along the

river. If Isabel is still standing in her original position, how far apart are Kenji and Isabel now? B. Isabel turns and walks 13 meters in the op-

posite direction, also along the river. About how far apart is Isabel from Kenji’s original position? Round to the nearest meter. 24.

Min is building a ramp from the ground to her front doorstep. The top of the doorstep is 3.5 feet above the ground, and the ramp is 9.5 feet long. How far will it extend from the house? Round to the nearest tenth of a foot.

25.

Hans leaned a 17-foot ladder against his house. He placed the base of the ladder 8 feet from his house, and the top of the ladder reached the base of his window. How high is the base of his window from the ground?

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Higher Roots The opposite of squaring is taking a square root. Similarly, the opposite of raising to any power n is taking the nth root. DEFINITION For all a and b, if an = b, and n is an integer greater than 1, a is the nth root of b.

NOTATION

n

__

The expression √b indicates the principal nth root of b when n is an even number.

PROPERTIES OF EVEN AND ODD ROOTS

n

__

Even roots have two real answers. If n is even and b is nonnegative, √b n __ indicates the principal, or positive, root and − √ b indicates the negative root. n

__

Odd roots have one real answer. If n is odd and b is positive or negative, √ b indicates the only root, which may be positive or negative.

For instance, 3 is the third root, or cube root, of 27 because 3 · 3 · 3 = 33 = 27. Just as 9 is a perfect square, 27 is a perfect cube. Although you cannot take the square root of a negative number, you can take an odd root of a negative number.____ The cube root of −27 is −3 because 3 (−3)3 = −27. This can be written as √−27 = −3.

Evaluating nth Roots Example 1 A.

4

Evaluate each radical expression.

___

√ 81

Solution Both 34 and (−3)4 equal 81, but the principal fourth root is 3, so ___ 4 √ 81 = 3. ■ B.

6

___

√ −1

Solution 5

A negative number does not have any even roots. ■

_______

√100,000 _______ 5 Solution √100,000 = 10 because 105 = 100,000. ■ _____ C.

D.

3

√ −125

Solution

366

UNIT 9

3

_____

√ −125 = −5 because (−5)3 = −125. ■

RATIONALS, IRRATIONALS, AND RADICALS

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Simplifying nth Roots

PROPERTIES If b is a perfect nth power, then b can be factored so that the exponent on each factor is n. If b is not a perfect nth power, then the nth root of b is irrational.

Variable expressions can also be perfect nth powers. Assume all variables are nonnegative.

PROPERTY For all a__and__b when n is odd, and for all nonnegative a and b when n is n n n ___ √ √ √ even, a · b = ab when n is an integer greater than 1.

Example 2 Simplify each expression. Assume all variables are nonnegative. 3

___

√ 54

A.

Solution 3

___

3

_____

√ 54 = √ 27 · 2 3

___

3

REMEMBER

__

= √27 · √2 3

__

3

Assuming __ a is positive when n is n even, √an = a.

__

= √33 · √2 3

__

= 3√2 ■ B.

___

____

√5y2 · √25y4

3

3

Solution ___

____

_____

√5y2 · √25y4 = √125y6

3

3

3

3

____

__

= √125 · √y6 3

__

3

____

= √53 · √(y2)3 3

= 5y2 ■

Simplifying Expressions with Fractional Exponents An exponent does not have to be an integer.

PROPERTY

1 __

n

__

n √ For all a when ___ n isnodd, __ m and for all nonnegative a when n is even, a = a n m __ m n and a = √ a = ( √a ) when n and m are integers greater than or equal to 1.

(continued)

HIGHER ROOTS

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Example 3

Simplify each expression.

1 __

A.

42

1 __

__

2

42 = √ 4 = 2 ■

Solution

2 __

B.

(−27)3

2 __

____

2

(−27)3 = ( √−27 ) = (−3)2 = 9 ■

Solution

3

1 __

−6254

C.

1 __

___

4

____

−6254 = −( √625 ) = −(5) = −5 ■

Solution

4

√ x14

D.

Solution ___ ____

4

4

x12x2 √x14 = √____

4

__

3 4 = √(x__ ) · √x2 4

4

= x3√x2 2 __

= x 3x 4 1 __

= x3x 2

__

= x3 √ x ■

Rationalizing Denominators with nth Roots When rationalizing a denominator with an nth root, be sure to multiply both the numerator and denominator by the radical expression that will make the radicand a perfect nth power. Example 4 Simplify.

REMEMBER

A.

An expression with a radical in the denominator is not simplified.

Solution 3__ ___ 3

√9

3__ ___ 3

√9

3

__

3 √__ 3__ ___ ___

=3

·3

√9

__

3

√3

3 3√ ___ = ____ 3 √ 27 __

3

3√3 ____

= 3 3

__

= √3 ■ B.

1 ____ ___ 5

√ 16

Solution 1 ____ ___ 5

√ 16

5

__

2 √__ 1 ___ ____ ___

=5

√ 16

·5

__

5

√2

2 √___ = ____ 5 √ 32 5

__

√2 ___

= 2 ■

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Problem Set Simplify. 1. 2. 3. 4.

3

___

√ 64

5

9.

6. 7. 8.

____

√ −64

____

__ 1

√ −32

10.

83

4__ ___

11.

162

4

√8

5 ____ ___

12.

3

√ 16

1 __

5.

6

(−512)3 10 _____ ____ 4

√ 125 3

4

4

____

√ 243

2562

18.

6 ____ ___

19.

7__ ___

20.

3 ____ ___

21.

18 _______ ______

______

√10,000 ____

√ −81

__ 1

14.

(−27)3

15.

643

16.

−2163

_____

√ −125

5

13.

1 __

22.

__ 2

1 __

17.

23.

5

√ 16

√3 3

√ 49

5

√10,000 3

5

25.

5

___

√x12

1 __

3

_____

√ 24x10

7

26. 27.

____

√ 486

Simplify. Assume all variables are nonnegative. 24.

___

√ 16

___

√x32

3

28.

______

5

____

5

______

√ 25x2 · √ 250x17

√ 192x11

Solve. 29.

The area of a square-shaped garden is 98 ft2. Find the length of each side of the garden. Express your answer in simplest radical form. The formula for the area of a square is A = s2.

*30.

Challenge The volume of a cube-shaped hat box is 24 in3. Find the length of each side of the hat box. Express your answer in simplest radical form. The formula for the volume of a cube is V = s3.

HIGHER ROOTS

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