ch6 ans

Prentice Hall Algebra 2 Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.

1

Name Class Date

6-1 Additional Vocabulary Support Roots and Radical ExpressionsComplete the vocabulary chart by fi lling in the missing information.

Word or Word Phrase

De nition Example

nth root Given the equation an 5 b, a is the nth root of b.

1.

radicand 2. Th e radicand in the expression !3 64 is 64.index Th e number that gives the degree

of the root.3.

cube root Th e third root of a number. 4.

principal root 5. Th e principal square root of 4 is 2.

Choose the word or phrase from the list that best completes each sentence.

cube root nth root radicand index principal root

6. Th e is the number under the radical sign in a radical expression.

7. Th e of 27 is 3.

8. Given the equation an 5 b, a is the of b.

9. In a radical expression, the indicates the degree of the root.

10. When a number has both a positive and a negative root, the positive root is

considered the .

34 5 81; 3 is the 4th root of 81.

The cube root of 8 is 2.

The index in the

expression 5!32 is 5.

The number under the radical sign.

The positive root when a number has both a positive and a negative root.

radicand

cube root

nth root

index

principal root


2

Name Class Date

6-1 Think About a PlanRoots and Radical ExpressionsBoat Building Boat builders share an old rule of thumb for sailboats. Th e maximum speed K in knots is 1.35 times the square root of the length L in feet of the boats waterline. a. A customer is planning to order a sailboat with a maximum speed

of 12 knots. How long should the waterline be? b. How much longer would the waterline have to be to achieve a maximum

speed of 15 knots?

1. Write an equation to relate the maximum speed K in knots to the length L in feet of a boats waterline.

2. How can you fi nd the length of a sailboats waterline if you know its maximum speed?

.

3. A customer is planning to order a sailboat with a maximum speed of 12 knots. How long should the waterline be?

4. How can you fi nd how much longer the waterline would have to be to achieve a maximum speed of 15 knots, compared to a maximum speed of 12 knots?

.

5. If a customer wants a sailboat with a maximum speed of 15 knots, how long should the waterline be?

6. How much longer would the waterline have to be to achieve a maximum speed of 15 knots?

K 5 1.35!L

Substitute the maximum speed for K and solve the resulting equation for L

Subtract the waterline length needed for a 12-knot maximum speed from the

waterline length needed for a 15-knot maximum speed

about 123 ft

about 44 ft

about 79 ft

Prentice Hall Gold Algebra 2 Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.

3

Name Class Date

6-1 Practice Form GRoots and Radical ExpressionsFind all the real square roots of each number.

1. 400 2. 2196 3. 10,000 4. 0.0625

Find all the real cube roots of each number.

5. 216 6. 2343 7. 20.064 8. 100027

Find all the real fourth roots of each number.

9. 281 10. 256 11. 0.0001 12. 625

Find each real root.

13. !144 14. 2!25 15. !20.01 16. !3 0.001 17. !4 0.0081 18. !3 27 19. !3 227 20. !0.09Simplify each radical expression. Use absolute value symbols when needed.

21. "81x4 22. "121y10 23. "3 8g6 24. "3 125x9 25. "5 243x5y15 26. "3 (x 2 9)3 27. "25(x 1 2)4 28. %3 64x9343 29. !3 20.008 30. %4 x481 31. "36x2y6 32. "4 (m 2 n)4 33. A cube has volume V 5 s3, where s is the length of a side. Find the side length

for a cube with volume 8000 cm3.

34. Th e temperature T in degrees Celsius (8C) of a liquid t minutes after heating is given by the formula T 5 8!t . When is the temperature 488C?

12

0.3

9x2

3xy3

6

220, 20

no real fourth roots

25

3

11 y5

x 2 9

20.2

6xy3x3

m 2 n

36 min

20 cm

27

no real square roots

24, 4

not a real number

23

2g2

5(x 1 2)2

20.4

2100, 100

20.1, 0.1

0.1

0.3

5x3

4x37

103

20.25, 0.25

25, 5


4

Name Class Date

Find the two real solutions of each equation.

35. x2 5 4 36. x4 5 81

37. x2 5 0.16 38. x2 5 1649

39. x4 5 16625 40. x2 5

121625

41. x2 5 0.000009 42. x4 5 0.0001

43. Th e number of new customers n that visit a dry cleaning shop in one year is directly related to the amount a (in dollars) spent on advertising. Th is relationship is represented by n3 5 13,824a. To attract 480 new customers, how much should the owners spend on advertising during the year?

44. Geometry Th e volume V of a sphere with radius r is given by the formula V 5 43 pr

3. a. What is the radius of a sphere with volume 36p cubic inches? b. If the volume increases by a factor of 8, what is the new radius?

45. A clothing manufacturer fi nds the number of defective blouses d is a function of the total number of blouses n produced at her factory. Th is function is d 5 0.000005n2.

a. What is the total number of blouses produced if 45 are defective? b. If the number of defective blouses increases by a factor of 9, how does the

total number of blouses change?

46. Th e velocity of a falling object can be found using the formula v2 5 64h, where v is the velocity (in feet per second) and h is the distance the object has already fallen.

a. What is the velocity of the object after a 10-foot fall? b. How much does the velocity increase if the object falls 20 feet

rather than 10 feet?

6-1 Practice (continued) Form GRoots and Radical Expressions

22, 2

$8000

3 in.

6 in.

3000

about 25.30 ft/sec

It has tripled.

about 10.48 ft/sec

20.003, 0.003

225, 25

23, 3

20.1, 0.1

20.4, 0.4

21125, 1125

247, 47

Prentice Hall Foundations Algebra 2 Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.

5

Name Class Date

Find all the real square roots of each number.

1. 625 2. 21.44 3. 1681

Find all the real cube roots of each number.

4. 2216 5. 164 6. 0.027

Find all the real fourth roots of each number.

7. 0.2401 8. 1 9. 21296

Find each real root. To start, fi nd a number whose square, cube, or fourth is equal to the radicand.

10. !400 11. 2!4 256 12. !3 2729 5 "(20)2

Simplify each radical expression. Use absolute value symbols when needed. To start, write the factors of the radicand as perfect squares, cubes, or fourths.

13. "25x6 14. "3 343x9y12 15. "4 16x16y20 5 "(5)2(x3)2

6-1 Practice Form KRoots and Radical Expressions

625

26

60.7

20

5x 3

no real roots

14

61

24

7x 3y 4

649

0.3

no real fourth roots

29

2x 4y 5


6

Name Class Date

16. Th e formula for the volume of a sphere is V 5 43 pr3. Solving for r, the radius of

a sphere is r 5 3 3V4p . If the volume of a sphere is 20 ft3, what is the radius of the sphere to the nearest hundredth?

Find the two real solutions of each equation.

17. x4 5 81 18. x2 5 144 19. x4 5 2401625

20. Writing Explain how you know whether or not to include the absolute value symbol on your root.

21. Arrange the numbers !3 264, 2!3 264, !64, and !6 64, in order from least to greatest.

22. Open-Ended Write a radical that has no real values.

23. Reasoning Th ere are no real nth roots of a number b. What can you conclude about the index n and the number b?

6-1 Practice (continued) Form KRoots and Radical Expressions

6 3 6 12 w 75

1.68 ft

If the index is odd, then you do not use the absolute value symbol on your root. If the index is even, then you need the absolute value symbol on those variable terms with an odd power.

Answers may vary. Sample: any even index radical with a negative radicand

The index n is even and the number b is negative.

3!264 , 6!64, 2 3!264 , !64


7

Name Class Date

Multiple Choice

For Exercises 16, choose the correct letter.

1. What is the real square root of 0.0064?

0.4 0.04

0.08 no real square root

2. What is the real cube root of 264?

4 28

24 no real cube root

3. What is the real fourth root of 2 1681?

23 249

223 no real fourth root

4. What is the value of !3 20.027? 20.3 0.3 20.03 0.03

5. What is the simplifi ed form of the expression "4x2y4? 2xy2 2 u x uy2 4xy2 2 u xy u

6. What are the real solutions of the equation x4 5 81?

29, 9 3 23, 3 23

Short Response

7. Th e volume V of a cube with side length s is V 5 s3. A cubical storage bin has volume 5832 cubic inches. What is the length of the side of the cube? Show your work.

6-1 Standardized Test Prep Roots and Radical Expressions

B

G

D

F

B

H

[2] V 5 s3, 5832 5 s3, s 53!5832 5 18; 18 in.

[1] incorrect side length OR no work shown[0] incorrect answer and no work shown OR no answer given


8

Name Class Date

Rounding Roots and RadicalsComputers treat radicals such as !2 as if they were rounded to a preassigned number of decimal places. Most computers round numbers according to an algorithm that uses the largest integer less than or equal to a given number. Th is function is called the greatest integer function and is written as y 5 fxg .

As you can see, the graph of the greatest integer function is not continuous. Th e open circles indicate that the endpoint is not included as part of the graph.

Th e command INT in most popular spreadsheet programs serves the same purpose as the greatest integer function. For instance, INT(3.84) 5 3; INT(21.99) 5 22; INT(7) 5 7.

To round a number x to r decimal places, a computer performs the following procedure:

Step 1 Multiply x by 10r.

Step 2 Add 0.5 to the result.

Step 3 Find INT of the result.

Step 4 Multiply the result by 102r.

Fill in the table below to see how this procedure works.

x r Step 1 Step 2 Step 3 Step 4

11.4825 3 11482.5 11483

132.718 2

34.999 1

A computer that rounds numbers after each operation may introduce rounding errors into calculations. To see the eff ects of rounding errors, perform each of the following computations for x 5 2 and diff erent r values. First fi nd the given root and write the answer to r 1 1 digits after the decimal. Carry out the four steps to get the answer and then raise the result to the given power. Write the answer again to r 1 1 digits after the decimal and carry out the four steps to get the fi nal answer.

x r Q!xR2 Q!3 x R32 6

2 3

2 1

6-1 EnrichmentRoots and Radical Expressions

O

2

2 2

x

y

2.000001

1.999

2.0

2.000000

2.000

2.2

11483

13271.8

349.99

13272

350

13272.3

350.49

132.72

35.0

11.483


9

Name Class Date

For any real numbers a and b and any positive integer n, if a raised to the nth power equals b, then a is an nth root of b. Use the radical sign to write a root. Th e following expressions are equivalent:

an 5 b g !n b 5 a

Problem

What are the real-number roots of each radical expression?

a. !3 343 Because (7)3 5 343, 7 is a third (cube) root of 343. Therefore, 3!343 5 7. (Notice that (27)3 5 2343, so 27 is not a cube root of 343.)

b. 4 1625 Because Q15R4 5 1625 and Q215R4 5 1625, both 15 and 2 15 are real-number fourth roots of 1625.

c. !3 20.064 Because (20.4)3 5 20.064,20.4 is a cube root of 20.064 and is, in fact, the only one. So, 3!20.064 5 20.4.

d. !225 Because (5)2 5 (25)2 5 25, neither 5 nor 25 are second (square) roots of 225. There are no real-number square roots of 225.

Exercises

Find the real-number roots of each radical expression.

1. !169 2. !3 729 3. !4 0.0016 4. 3 2 18 5. 4121 6. 3 125216 7. 2 425 8. !4 0.1296 9. !3 20.343 10. !4 20.0001 11. 5 1243 12. 3 8125

6-1 Reteaching Roots and Radical Expressions

an 5 g !n!!b!b 5 a!power index radicand

radical sign

213, 13

212

no real sq root

no real 4th root

9

2 211, 2

11

20.6, 0.6

13

20.2, 0.2

56

20.7

25


10

Name Class Date

You cannot assume that "n an 5 a. For example, "(26)2 5 !36 5 6, not 26. Th is leads to the following property for any real number a:

If n is odd "n an 5 a If n is even "n an 5 u a u

Problem

What is the simplifi ed form of each radical expression?

a. "3 1000x3y9 "3 1000x3y9 5 "3 103x3(y3)3 Write each factor as a cube. 5 "3 (10xy3)3 Write as the cube of a product. 5 10xy3 Simplify.

b. 4 256g8

h4k16

Write each factor as a power of 4.

5 4 a4g2

hk4b4 Write as the fourth power of a quotient.

54g2

u h u k4 Simplify.

Th e absolute value symbols are needed to ensure the root is positive when h is negative. Note that 4g2 and k4 are never negative.

Exercises

Simplify each radical expression. Use absolute value symbols when needed.

13. "36x2 14. "3 216y3 15. 1100x2 16.

"x20"y8 17. 3 (x 1 3)3

(x 2 4)6 18. "5 x10y15z5

19. 3 27z3

(z 1 12)6 20. "4 2401x12 21. 3 1331x3

22. 4 (y 2 4)8

(z 1 9)4 23. 3 a

6b6

c3 24. "3 2x3y6

6-1 Reteaching (continued) Roots and Radical Expressions

4 44(g2)4

h4(k4)454 256g

8

h4k16

6x

x10

y4

3z(z 1 12)2

(y 2 4)2

z 1 9

6y

x 1 3(x 2 4)2

7x3

a2b2c

110x

x2y3z

11x

2xy2


11

Name Class Date

6-2 Additional Vocabulary Support Multiplying and Dividing Radical ExpressionsCombining Radicals: Products

If !n a and !n b are real numbers, then !n a ? !n b 5 !n ab.Sample !3 8 ? !3 27 5 !3 8 ? 27 5 !3 216 5 6

Solve.

1. !3 16 ? !3 4 5 2. Which of the following products can be simplifi ed? Circle the correct answer.

!3 12 ? !6 !4 16 ? !4 24 !4 35 ? !3 10 3. Write the radical expression !3 32x4 in simplest form. 4. Which of the following products cannot be simplifi ed? Circle the correct answer.

!4 15 ? !4 4 !4 ? !12 !4 10 ? !3 5 5. "4x2y3 ? "27x2y2 5

Combining Radicals: Quotients

If !n a and !n b are real numbers and b 2 0, then !n a!n b 5 nab .Sample !8!2 5 82 5 !4 5 2

Solve.

6. Which of the following quotients can be simplifi ed? Circle the correct answer.

3 !12 3!4 !

3 6"3 !4 20!3 15

7. Write the radical expression "64x4"4x2 in simplest form. 8. Rewriting an expression so that there are no radicals in any denominator and no

denominators in any radical is called .

3!16 ? 4 5 3!64 5 4

3!4x

6x2y2"3y

l4xl

rationalizing the denominator

2x


12

Name Class Date

6-2 Think About a Plan Multiplying and Dividing Radical ExpressionsSatellites Th e circular velocity v, in miles per hour of a satellite orbiting Earth is given by the formula v 5 1.24 3 10

12

r , where r is the distance in miles from the satellite to the center of the Earth. How much greater is the velocity of a satellite orbiting at an altitude of 100 mi than the velocity of a satellite orbiting at an altitude of 200 mi? (Th e radius of the Earth is 3950 mi.)

Know

1. Th e fi rst satellite orbits at an altitude of zz.

2. Th e second satellite orbits at an altitude of zz.

3. Th e distance from the surface of the Earth to its center is zz.

Need

4. To solve the problem I need to fi nd:

.

Plan

5. Rewrite the formula for the circular velocity of a satellite using a for the altitude of the satellite.

6. Use your formula to fi nd the velocity of a satellite orbiting at an altitude of 100 mi.

7. Use your formula to fi nd the velocity of a satellite orbiting at an altitude of 200 mi.

8. How much greater is the velocity of a satellite orbiting at an altitude of 100 mi than one orbiting at an altitude of 200 mi?

100 mi

200 mi

the difference in the velocities of a satellite orbiting at an altitude of 100 mi

and one orbiting at an altitude of 200 mi

v 5 1.24 3 1012

a 1 3950

about 17,498 mi/h

about 17,286 mi/h

about 212 mi/h

3950 mi


13

Name Class Date

Multiply, if possible. Th en simplify.

1. !4 ? !25 2. !81 ? !36 3. !3 ? !3 27 4. "3 45 ? "3 75 5. !18 ? !50 6. !3 216 ? !3 4Simplify. Assume that all variables are positive.

7. "36x3 8. "3 125y2z4 9. "18k610. "3 216a12 11. "x2y10z 12. "4 256s7t12 13. "3 216x4y3 14. "75r3 15. "4 625u5v8Multiply and simplify. Assume that all variables are positive.

16. !4 ? !6 17. "9x2 ? "9y5 18. "3 50x2z5 ? "3 15y3z 19. 4!2x ? 3!8x 20. !xy ? !4xy 21. 9!2 ? 3!y 22. "12x2y ? "3xy4 23. "3 29x2y4 ? "3 12xy 24. 7"3y2 ? 2"6x3yDivide and simplify. Assume that all variables are positive.

25. "75"3 26. "63xy

3

"7y 27. "54x5y3"2x2y

28. "6x"3x 29.

3"4x23"x 30. 4243k

3

3k7

31. "(2x)2"(5y)4 32.

3"18y23"12y 33. 162a6a3

6-2 Practice Form GMultiplying and Dividing Radical Expressions

15 30 24

10

2"6

6xy2"xy 23xy 3"4y2 42xy"2xy

54

9xy2!y

9

5yz23"6x2

6x!x 5z 3"y2z 3k3"222a4 3!2 xy5"z 4st3 4"s36xy 3"x 5r"3r 5uv2 4!u

48x 2xy 27!2y

5 3y!x 3xy!3x

3"3a

"2 3"4x 3k2x

25y2

3"12y2


14

Name Class Date

Rationalize the denominator of each expression. Assume that all variables are positive.

34. !y!5 35. "18x

2y"2y3 36. "3 7xy2"3 4x2

37. 9x2 38. !xy!3x 39. 3 x23y

40. !4 2x"4 3x2 41. x8y 42. 3 3a4b2c

43. What is the area of a rectangle with length !175 in. and width !63 in.? 44. Th e area of a rectangle is 30 m2. If the length is !75 m, what is the width? 45. Th e volume of a right circular cone is V 5 13pr

2h, where r is the radius of the base and h is the height of the cone. Solve the formula for r. Rationalize the denominator.

46. Th e volume of a sphere of radius r isV 5 43pr3.

a. Use the formula to fi nd r in terms of V. Rationalize the denominator. b. Use your answer to part (a) to fi nd the radius of a sphere with volume

100 cubic inches. Round to the nearest hundredth.

Simplify each expression. Rationalize all denominators. Assume that all variables are positive.

47. !14 ? !21 48. !3 150 ? !3 20 49. !3Q!12 2 !6R 50.

6!2x5!3 51. 8"3 2x2 52. 5!

3 xy4"3 25xy2

6-2 Practice (continued) Form GMultiplying and Dividing Radical Expressions

"5y5

3xy

3"14x2y22x

3"2x2

4"54x33x

"3y3

"2xy4y

3"9x2y23y

3"6abc22bc

105 in.2

2"3 m

r 5 "3hVh

2.88 in.

7"6 6 2 3"22"6x

54

3"4xx

3"5y210

3"3

r 5 3"3V4

; r 53"62V

2


15

Name Class Date

6-2 Practice Form KMultiplying and Dividing Radical ExpressionsMultiply, if possible. Th en simplify. To start, identify the index of each radical.

1. !3 4 ? !3 6 2. !5 ? !8 3. !3 6 ? !4 9index of both radicals is 3!3 4 ? 6

Simplify. Assume all variables are positive. To start, change the radicand to factors with the necessary exponent.

4. "3 27x6 5. "48x3y4 6. "5 128x2y255 "3 33 ? (x2)3

Multiply and simplify. Assume all variables are positive.

7. !12 ? !3 8. "4 7x6 ? "4 32x2 9. 2"3 6x4y ? 3"3 9x5y2

Simplify each expression. Assume all variables are positive.

10. !3 4 ? !3 80 11. 5"2xy6 ? 2"2x3y 12. !5Q!5 1 !15R

13. Error Analysis Your classmate simplifi ed "5x3 ? "3 5xy2 to 5x2y. What mistake did she make? What is the correct answer?

14. A square rug has sides measuring !3 16 ft by !3 16 ft. What is the area of the rug?

2 3 !3

3x 2

6

4 3!5

They are different, so you cannot multiply the radicands.

4 3!4 ft 2

2!10

4xy 2!3x

2x 2 4!14

20x 2y 3!y

The indexes are different, so you cannot multiply.

2y 5 5"4x 2

18x 3y 3!2

5 1 5!3

She thought the indexes were the same.


16

Name Class Date

Divide and simplify. Assume all variables are positive. To start, write the quotient of roots as a root of a quotient.

15. "36x6"9x4 16.

"4 405x8y2"4 5x3y2 17. "3 75x7y2"3 25x4

5 36x6

9x4

Rationalize the denominator of each quotient. Assume all variables are positive. To start, multiply the numerator and denominator by the appropriate radical expression to eliminate the radical.

18. !26!3 19. !

3 x!3 2 20. "7x4y!5xy

5 !26!3 ? !3!3

21. Einsteins famous formula E 5 mc2 relates energy E, mass m, and the speed of light c. Solve the formula for c. Rationalize the denominator.

22. Th e formula h 5 16t2 is used to measure the time t it takes for an object to free fall from height h. If an object falls from a height of h 5 18a5 ft, how long did it take for the object to fall in terms of a?

6-2 Practice (continued) Form KMultiplying and Dividing Radical Expressions

2x

!783

c 5 Em; c 5 !Emm

3a 2!2a4 seconds

3x 4!x

3!4x2

x 3"3y 2

x !35x5


17

Name Class Date

6-2 Standardized Test Prep Multiplying and Dividing Radical ExpressionsMultiple Choice

For Exercises 15, choose the correct letter. Assume that all variables are positive.

1. What is the simplest form of !3 249x ? "3 7x2? 7x!7x 27x 7x 27"3 x2

2. What is the simplest form of "80x7y6? 2x3y3!20x 4x6y6"5x3 4"5x7y6 4x3y3!5x

3. What is the simplest form of "3 25xy2 ? "3 15x2? 5x"3 3y2 5x!3 3y 15xy!3 y 5xy!15x

4. What is the simplest form of "75x5"12xy2? 5"3x4

2"3y2 5x2

2y 5x!x

2y 5x2y

2

5. What is the simplest form of 2"3 x2y"3 4xy2?

"3 x2y

2y x!3 2y

y "3 2xy2

y !3 2yxy

Short Response

6. Th e volume V of a wooden beam is V 5 ls2, where l is the length of the beam and s is the length of one side of its square cross section. If the volume of the beam is 1200 in.3 and its length is 96 in., what is the side length? Show your work.

B

I

A

G

C

[2] V 5 ls2; s 5 Vl 5 120096 5 !12.5 N 3.5 in.[1] appropriate methods but with computational errors OR correct answer without

work shown[0] incorrect answer and no work shown OR no answer given


18

Name Class Date

6-2 EnrichmentMultiplying and Dividing Radical ExpressionsTo simplify the radical !n a, you look for a perfect nth power among the factors of the radicand a. When this factor is not obvious, it is helpful to factor the number into primes. Prime numbers are important in many aspects of mathematics. Several mathematicians throughout history have unsuccessfully tried to fi nd a pattern that would generate the nth prime number. Other mathematicians have off ered conjectures about primes that remain unresolved.

1. Goldbachs Conjecture states that every even number n . 2 can be written as the sum of two primes. For example, 4 5 2 1 2 and 10 5 3 1 7. Choose three even numbers larger than 50 and write them as a sum of two primes.

2. Th e Odd Goldbachs Conjecture states that every odd number n . 5 can be written as the sum of three primes. For example, 7 5 2 1 2 1 3. Choose three odd numbers larger than 50 and write them as the sum of three primes.

3. Another interesting pattern emerges when you examine a subset of the prime numbers. Make a list of the primes less than 50.

4. Make this list smaller by eliminating 2 and all primes that are 1 less than a multiple of 4.

5. Th e remaining primes in the list above are related in an interesting way. You can write each prime as the sum of two squares. Express each of these primes as a sum of two squares.

6. A Cullen number, named after an Irish mathematician James Cullen, is a natural number of the form n 3 2n 1 1. Determine the fi rst four Cullen numbers. Th at is, let n 5 1, 2, 3, 4.

7. What is the smallest Cullen number that is a prime number? (Th e next Cullen number that is a prime occurs when n 5 141!)

8. A palindrome is a number that reads the same forward and backward. For example, 121 is a palindromic number. List the seven palindromic primes that are less than 140.

Answers may vary. Sample: 52 5 47 1 5.

Answers may vary. Sample: 51 5 37 1 11 1 3.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

5, 13, 17, 29, 37, 41

5 5 1 1 4, 13 5 4 1 9, 17 5 1 1 16, 29 5 4 1 25, 37 5 1 1 36, 41 5 16 1 25

3, 9, 25, 65

3 when n 5 1

2, 3, 5, 7, 11, 101, 131


19

Name Class Date

6-2 ReteachingMultiplying and Dividing Radical ExpressionsYou can simplify a radical if the radicand has a factor that is a perfect nth power and n is the index of the radical. For example:

!n xynz 5 y!n xz Problem

What is the simplest form of each product?

a. !3 12 ? !3 10 !3 12 ? !3 10 5 !3 12 ? 10 Use 5 "3 22 ? 3 ? 2 ? 5 Write as a product of factors. 5 "3 23 ? 3 ? 5 Find perfect third powers. 5 "3 23 ? "3 3 ? 5 Use n!ab 5 n !a ? n!b. 5 2!3 15 Use n"an 5 a to simplify.b. "7xy3 ? "21xy2 "7xy3 ? "21xy2 5 "7xy3 ? 21xy2 Use n!a ? n!b 5 n !ab. 5 "7xy2y ? 3 ? 7xy2 Write as a product of factors. 5 "72x2(y2)2 ? 3y Find perfect second powers. 5 7xy2"3y Use n!an 5 a to simplify.Exercises

Simplify each product.

1. !15x ? !35x 2. "3 50y2 ? "3 20y 3. "3 36x2y5 ? "3 26x2y 4. 5"7x3y ? "28y2 5. 2"3 9x5y2 ? "3 2x2y5 6. !3 Q!12 2 !21 R

n!a ? n!b 5 n !ab.

5x"21 10y 26xy2 3!x70xy"xy 2x2y2 3!18xy 6 2 3"7


20

Name Class Date

Rationalizing the denominator means that you are rewriting the expression so that no radicals appear in the denominator and there are no fractions inside the radical.

Problem

What is the simplest form of !9y!2x?

Rationalize the denominator and simplify. Assume that all variables are positive.

!9y!2x 5 9y2x Rewrite as a square root of a fraction.

5 9y ? 2x2x ? 2x Make the denominator a perfect square. 5 18xy4x2 Simplify. 5

!18xy"22 ? x2 Write the denominator as a product of perfect squares.

5"18xy

2x Simplify the denominator.

5"32 ? 2 ? x ? y

2x Simplify the numerator.

53"2xy

2x Use n!an 5 a to simplify.

Exercises

Rationalize the denominator of each expression. Assume that all variables are positive.

7. !5!x 8. "

3 6ab2"3 2a4b 9. "4 9y"4 x 10. "10xy

3

"12y2 11.

4"3 k916"3 k5 12. 3x

5

5y 13. "4 10"4 z2 14. 3 19a

2babc4

6-2 Reteaching (continued)Multiplying and Dividing Radical Expressions

"5xx

3"3ba

4"9x3yx

"30xy6

k 3"k4

x2"15xy5y

4"10z2z

3"19ac2c2


21

Name Class Date

Th e column on the left shows the steps used to rationalize a denominator. Use the column on the left to answer each question in the column on the right.

Problem Rationalizing the Denominator

Write the expression 4!3!7 1 !3 with a rationalized denominator.

1. What does it mean to rationalize a denominator?

Multiply the numerator and the denominator by the conjugate of the denominator.

4!3!7 1 !3 ? !7 2 !3!7 2 !3 2. What are conjugates?

The radicals in the denominator cancel out.4!3A!7 2 !3B

7 2 3

3. Write and solve an equation to show why the radicals in the denominator cancel out.

Distribute !3 in the numerator.4(!3 ? !7 2 !3 ? !3)

7 2 3

4. What property allows you to distribute the !3?

Simplify.4(!21 2 3)

4

5. Why do the fours in the numerator and the denominator cancel out?

Simplify.

!21 2 3 6. What number multiplied by !21 would produce a product of 21?

6-3 Additional Vocabulary Support Binomial Radical Expressions

Sample answer: It means to write

an expression so that there are no

radicals in any denominators and no

denominators in any radicals.

Conjugates are expressions that differ

only in the signs of the rst or second

terms.

(!7 1 !3)(!7 2 !3) 5(!7 ? !7) 2 (!7 ? !3) 1(!7 ? !3) 2 (!3 ? !3) 5 7 2 3

The Distributive Property

Sample answer: Because 4 divided by

4 equals 1.

!21


22

Name Class Date

6-3 Think About a PlanBinomial Radical ExpressionsGeometry Show that the right triangle with legs of length !2 2 1 and !2 1 1 is similar to the right triangle with legs of length 6 2 !32 and 2.Understanding the Problem

1. What is the length of the shortest leg of the fi rst triangle? Explain.

2. What is the length of the shortest leg of the second triangle? Explain.

3. Which legs in the two triangles are corresponding legs?

Planning the Solution

4. Write a proportion that can be used to show that the two triangles are similar.

Getting an Answer

5. Simplify your proportion to show that the two triangles are similar.

!2 2 1 ; because !2 5 !2, !2 2 1 must be less than !2 1 1

6 2 !32; because !32 is between 5 and 6, 6 2 !32 must be between 0 and 1,which is less than 2.

The smaller leg in the rst triangle corresponds to the smaller leg in the second

triangle. The larger leg in the rst triangle corresponds to the larger leg in the

second triangle.

!2 2 1!2 1 1 0 6 2 !322

!2 2 1!2 1 1 0 6 2 !322 2(!2 2 1) 0 (!2 1 1)(6 2 !32)

2!2 2 2 0 6!2 2 !64 1 6 2 !32 2!2 2 2 0 6!2 2 8 1 6 2 4!2 2!2 2 2 5 2!2 2 2


23

Name Class Date

6-3 Practice Form GBinomial Radical ExpressionsAdd or subtract if possible.

1. 9!3 1 2!3 2. 5!2 1 2!3 3. 3!7 2 7 3!x 4. 14!3 xy 2 3!3 xy 5. 8!3 x 1 2!3 y 6. 5!3 xy 1 !3 xy 7. !3x 2 2!3x 8. 6!2 2 5!3 2 9. 7!x 1 x!7Simplify.

10. 3!32 1 2!50 11. !200 2 !72 12. !3 81 2 3!3 3 13. 2!4 48 1 3!4 243 14. 3!75 1 2!12 15. !3 250 2 !3 54 16. !28 2 !63 17. 3!4 32 2 2!4 162 18. !125 2 2!20Multiply.

19. A1 2 !5B A2 2 !5B 20. A1 1 4!10B A2 2 !10B 21. A1 2 3!7B A4 2 3!7B 22. (4 2 2!3)2 23. (!2 1 !7)2 24. A2!3 2 3!2B2 25. A4 2 !3B A2 1 !3B 26. A3 1 !11B A4 2 !11B 27. A3!2 2 2!3B2Multiply each pair of conjugates.

28. (3!2 2 9)(3!2 1 9) 29. (1 2 !7)(1 1 !7) 30. (5!3 1 !2)(5!3 2 !2) 31. (3!2 2 2!3)(3!2 1 2!3) 32. (!11 1 5)(!11 2 5) 33. (2!7 1 3!3)(2!7 2 3!3)

8 3!x 1 2 3!y

6!2 2 5 3!2

4!2

19!3

0

238 1 7"10

5!2 1 2!3

9 1 2!141 1 !11

2!3x

22!2

13 4!32!7

7 2 3!5

11!3

28 2 16!3

11 3!xy

5 1 2!3

263

73

214

26

6

1

6 3!xy

7!x 1 x!7

0

2 3!2!5

67 2 15!7

3!7 2 7 3!x

30 2 12!630 2 12!6


24

Name Class Date

6-3 Practice (continued) Form GBinomial Radical ExpressionsRationalize each denominator. Simplify the answer.

34. 3 2 !10!5 2 !2 35. 2 1 !14!7 1 !2 36. 2 1 !

3 x!3 xSimplify. Assume that all the variables are positive.

37. !28 1 4!63 2 2!7 38. 6!40 2 2!90 2 3!160 39. 3!12 1 7!75 2 !54 40. 4!3 81 1 2!3 72 2 3!3 24 41. 3!225x 1 5!144x 42. 6"45y2 1 4"20y2 43. A3!y 2 !5B A2!y 1 5!5B 44. A!x 2 !3B A!x 1 !3B 45. A park in the shape of a triangle has a sidewalk dividing it into two parts.

a. If a man walks around the perimeter of the park, how far will he walk? b. What is the area of the park?

46. Th e area of a rectangle is 10 in.2. Th e length is A2 1 !2B in. What is the width? 47. One solution to the equation x2 1 2x 2 2 5 0 is 21 1 !3. To show this,

let x 5 21 1 !3 and answer each of the following questions. a. What is x2? b. What is 2x? c. Using your answers to parts (a) and (b), what is the sum x2 1 2x 2 2?

600 ft

300 ft

300 6 ft

side

wal

k V

300 3 ftV

300

3

ftV

5(2 2 !2) in.

(900 1 300!3 1 300!6) ft or about 2154 ft270,000 1 90,000!3

2 ft2 or about 212,942 ft2

4 2 2!322 1 2!3

0

!5 2 2!23

12!741!3 2 3!6

105!x6y 1 13!5y 2 25 x 2 3

26y!56 3!3 1 4 3!926!10

!2 x 1 2 3"x2x


25

Name Class Date

Simplify if possible. To start, determine if the expressions contain like radicals.

1. 3!5 1 4!5 2. 8!3 4 2 6!3 4 3. 2!xy 1 2!yboth radicals

4. A fl oor tile is made up of smaller squares. Each square measures 3 in. on each side. Find the perimeter of the fl oor tile.

Simplify. To start, factor each radicand.

5. !18 1 !32 6. !4 324 2 !4 2500 7. !3 192 1 !3 245 !9 ? 2 1 !16 ? 2

Multiply.

8. A3 2 !6B A2 2 !6B 9. A5 1 !5B A1 2 !5B 10. A4 1 !7B2

Multiply each pair of conjugates.

11. A7 2 !2B A7 1 !2B 12. A1 1 3!3B A1 2 3!3B 13. A6 1 4!7B A6 2 4!7B

6-3 Practice Form KBinomial Radical Expressions

7!5

7!2

12 2 5!6

47

2 3!4

22 4!4 or 22!2

24!5

226

no; cannot simplify

6 3!3

23 1 8"7

276

24!2 in.


26

Name Class Date

Rationalize each denominator. Simplify the answer.

14. 32 1 !6 15. 7 1 !56 2 !5 16. 1 2 2!104 1 !105 3

2 1 !6 ? 2 2 !62 2 !6

17. A section of mosaic tile wall has the design shown at the right. Th e design is made up of equilateral triangles. Each side of the large triangle is 4 in. and each side of a small triangle is 2 in. Find the total area of the design to the nearest tenth of an inch.

Simplify. Assume that all variables are positive.

18. !45 2 !80 1 !245 19. A2 2 !98B A3 1 !18B 20. 6"192xy2 1 4"3xy2

21. Error Analysis A classmate simplifi ed the

expression 11 2 !2 using the steps shown.

What mistake did your classmate make?

What is the correct answer?

22. Writing Explain the fi rst step in simplifying !405 1 !80 2 !5.

6-3 Practice (continued) Form KBinomial Radical Expressions

11 2 !2 ? 1 2 !21 2 !25 1 2 !21 2 2 5 1 2 !221 5 21 1 !2

23 1 32!6

6!5

47 1 13!531

236 2 15!2

4 2 32!10

52y!3x

A N 17.3 in.2

The student multiplied the denominator by itself instead of by its conjugate; 21 2 !2

First, factor each radicand so you can combine like radicals.


27

Name Class Date

Multiple Choice


1. What is the simplest form of 2!72 2 3!32? 2!72 2 3!32 24!2 22!2 0

2. What is the simplest form of A2 2 !7B A1 1 2!7B? 212 1 3!7 16 1 5!7 212 2 3!7 3 1 !7

3. What is the simplest form of A!2 1 !7B A!2 2 !7B? 9 1 2!14 9 2 2!14 25 9

4. What is the simplest form of 72 1 !5?

214 1 7!5 214 2 7!5 14 1 7!5 14 2 7!5

5. What is the simplest form of 8!3 5 2 !3 40 2 2!3 135? 16!3 5 12!3 5 4!3 5 0

Short Response

6. A hiker drops a rock from the rim of the Grand Canyon. Th e distance it falls d in feet after t seconds is given by the function d 5 16t2. How far has the rock fallen after (3 1 !2) seconds? Show your work.

6-3 Standardized Test PrepBinomial Radical Expressions

D

F

C

F

D

[2] d 5 16t2 5 16(3 1 !2)2 5 16(11 1 6!2) 5 176 1 96!2 ft[1] appropriate method but with computational errors[0] incorrect answer and no work shown OR no answer given


28

Name Class Date

Consider how you might use a calculator to fi nd the square of negative three. If you enter the expression 232, your calculator produces an answer of 29. However, the square of negative three is (23)2 5 (23)(23) 5 9. Calculators follow the order of operations. Th erefore, a calculator will compute 232 as the opposite of 32. Th e correct input is (23)2, which is correctly evaluated as 9. Be sure to follow the order of operations when expanding binomial radical expressions.

1. Consider the algebraic expression (a 1 b)2. Is (a 1 b)2 equivalent to a2 1 b2? If yes, explain. If not, explain why it is not mathematically logical and give a counterexample.

2. Are there values of a and b for which (a 1 b)2 5 a2 1 b2?

Consider each pair of expressions below for nonnegative values of the variables. State whether they are equivalent expressions. If yes, explain. If not, give a counterexample.

3. "x2 1 y2, "x2 1 "y2

4. "ab , ab

5. Q!aR2, a

6. Q"x2 1 y2R2, x 1 y

6-3 EnrichmentBinomial Radical Expressions

Answer may vary. Sample: (a 1 b)2 means (a 1 b)(a 1 b) which, when expanded, is a2 1 2ab 1 b2, which is not equivalent to a2 1 b2.

Answers may vary. Sample: a 5 1, b 5 0

Answers may vary. Sample: These expressions are not equivalent. Let x 5 2 and y 5 3 then "22 1 32 5"13 u"4 1 "9

Answers may vary. Sample: These expressions are not equivalent. Let a = 6 and

b 5 2 then !62 N 1.22 and 62 5 !3 N 1.73

Answers may vary. Sample: These expressions are equivalent. A!aB2 5 A!aB A!aB 5"a2 5 a for all a L 0.

Answers may vary. Sample: These expressions are not equivalent. Q"x2 1 y2RQ"x2 1 y2R 5"(x2 1 y2)2 5 x2 1 y2


29

Name Class Date

Two radical expressions are like radicals if they have the same index and the same radicand.

Compare radical expressions to the terms in a polynomial expression.

Like terms: 4x3 11x3 Th e power and the variable are the same

Unlike terms: 4y3 11x3 4y2 Either the power or the variable are not the same.

Like radicals: 4!3 6 11!3 6 Th e index and the radicand are the sameUnlike radicals: 4!3 5 11!3 6 4 2"6 Either the index or the radicand are not the same.When adding or subtracting radical expressions, simplify each radical so that you can fi nd like radicals.

Problem

What is the sum? !63 1 !28 !63 1 !28 5 !9 ? 7 1 !4 ? 7 Factor each radicand. 5 "32 ? 7 1 "22 ? 7 Find perfect squares. 5 "32"7 1 "22"7 Use n!ab 5 n !a ? n!b. 5 3!7 1 2!7 Use n"an 5 a to simplify. 5 5!7 Add like radicals.Th e sum is 5!7.Exercises

Simplify.

1. !150 2 !24 2. !3 135 1 !3 40 3. 6!3 2 !75 4. 5!3 2 2 !3 54 5. 2!48 1 !147 2 !27 6. 8!3 3x 2 !3 24x 1 !3 192x

6-3 ReteachingBinomial Radical Expressions

3!6 !3

10 3!3x2 3!2

5 3!50


30

Name Class Date

Conjugates, such as !a 1 !b and !a 2 !b, diff er only in the sign of the second term. If a and b are rational numbers, then the product of conjugates produce a rational number:

Q!a 1 !bRQ!a 2 !b R 5 Q!a R2 2 Q!b R2 5 a 2 b. You can use the conjugate of a radical denominator to rationalize the

denominator.

Problem

What is the product? Q2!7 2 !5RQ2!7 1 !5R Q2!7 2 !5RQ2!7 1 !5R These are conjugates. 5 Q2!7R2 2 Q!5R2 Use the difference of squares formula. 5 28 2 5 5 23 Simplify.

Problem

How can you write the expression with a rationalized denominator? 4!21 1 !3

4!2

1 1 !3 5

4!21 1 !3 ? 1 2 !31 2 !3 Use the conjugate of 1 1 !3 to rationalize the denominator.

54!2 2 4!6

1 2 3 Multiply.

54!2 2 4!6

22 5 2A4!2 2 4!6B

2 Simplify.

524!2 1 4!6

2 5 22!2 1 2!6Exercises

Simplify. Rationalize all denominators.

7. A3 1 !6B A3 2 !6B 8. 2!3 1 15 2 !3 9. Q4!6 2 1RQ!6 1 4R

10. 2 2 !72 1 !7 11. A2!8 2 6B A!8 2 4B 12. !52 1 !3

6-3 Reteaching (continued)Binomial Radical Expressions

3 "3 1 12

20 1 15!640 2 28!2 2!5 2 !15211 1 4"7

3


31

Name Class Date

6-4 Additional Vocabulary Support Rational ExponentsChoose the word or phrase from the list that best matches each sentence.

rational exponent radical form exponential form

1. Th e expression "4 y3 is written in . 2. A is an exponent written in fractional form.

3. Th e expression x 35 is written in .

Write each expression in exponential form.

4. "4 y7 5 5. (!3 x)4 5 6. (!5 a)3 5 7. !8 r 5 Write each expression in radical form.

8. w34 5

9. b52 5

10. h12 5

11. g 37 5

Multiple Choice

12. What is "6 y4!3 y in simplest terms?

y12 !3 y "3 y4 y 23

radical form

rational exponent

exponential form

y 74

x 43

a35

4"w3

7"g3

"b5"h

r 18

B


32

Name Class Date

6-4 Think About a PlanRational ExponentsScience A desktop world globe has a volume of about 1386 cubic inches. Th e radius of the Earth is approximately equal to the radius of the globe raised to the 10th power. Find the radius of the Earth. (Hint: Use the formula V 5 43pr

3 for the volume of a sphere.)

Know

1. Th e volume of the globe is zz.

2. Th e radius of the Earth is equal to .

Need

3. To solve the problem I need to fi nd .

Plan

4. Write an equation relating the radius of the globe rG to the radius of the Earth rE.

5. How can you represent the radius of the globe in terms of the radius of the Earth?

6. Write an equation to represent the volume of the globe.

7. Use your previous equation and your equation from Exercise 5 to write an equation to fi nd the radius of the Earth.

8. Solve your equation to fi nd the radius of the Earth.

1386 in.3

the radius of the globe raised to the 10th power

the radius of the Earth

about 251,000,000 in. or 3961 mi

rE 5 rG10

rG 5 r1

10E

1386 5 43r3

G

1386 5 43 r 110E 3

Name Class Date


33

6-4 Practice Form GRational ExponentsSimplify each expression.

1. 12513 2. 64

12 3. 32

15

4. 712 ? 7

12 5. (25)

13 ? (25)

13 ? (25)

13 6. 3

12 ? 75

12

7. 1113 ? 11

13 ? 11

13 8. 7

12 ? 28

12 9. 8

14 ? 32

14

10. 1212 ? 27

12 11. 12

13 ? 45

13 ? 50

13 12. 18

12 ? 98

12

Write each expression in radical form.

13. x43 14. (2y)

13 15. a1.5

16. b15 17. z

23 18. (ab)

14

19. m2.4 20. t227 21. a21.6


22. "x3 23. !3 m 24. !5y 25. "3 2y2 26. Q!4 bR3 27. !26 28. "(6a)4 29. "5 n4 30. "4 (5ab)3 31. Th e rate of infl ation i that raises the cost of an item from the present value P to

the future value F over t years is found using the formula i 5 QFPR1t 2 1. Round your answers to the nearest tenth of a percent.

a. What is the rate of infl ation for which a television set costing $1000 today will become one costing $1500 in 3 years?

b. What is the rate of infl ation that will result in the price P doubling (that is, F 5 2P) in 10 years?

5 8 2

7 25

1411

18

4

15

4230

14.5%

7.2%

3"x45!b

5"m12 17"t2 15"a83"z2 4!ab

3"2y "a3

x 32 m

13 (5y)

12

213y

23

36a2 n45 (5ab)

34

b34 (26)

12

Name Class Date


34

6-4 Practice (continued) Form GRational ExponentsWrite each expression in simplest form. Assume that all variables are positive.

32. Q8114R 4 33. Q3215R5 34. A2564B14 35. 70 36. 8

23 37. (227)

23

38. x12 ? x

13 39. 2y

12 ? y 40. A82B13

41. 3.60 42. Q 116R14 43. Q278 R23 44. "8 0 45. Q3x12R Q4x23R 46. 12y13

4y12

47. Q3a12 b13R2 48. Qy23R29 49. Qa23b212R26 50. y

25 ? y

38 51. ax47

x23

b 52. Q2a14R3 53. 812

12 54. Q2x25RQ6x14R 55. Q9x4y22R12

56. a27x664y4

b13 57. x12 y23x

13 y

12

58. y58 4 y

12

59. x14 ? x

16 ? x

13 60. a x213 y

x23 y2

12

b2 61. a 12x875y10

b12 62. In a test kitchen, researchers have measured the radius of a ball of dough

made with a new quick-acting yeast. Based on their data, the radius r of the dough ball, in centimeters, is given by r 5 5(1.05)

t3 after t minutes. Round the

answers to the following questions to the nearest tenth of a cm. a. What is the radius after 5 minutes? b. What is the radius after 20 minutes? c. What is the radius after 43 minutes?

32

4

1

x 2

21

3

y 16

12x 1320

1y6

12x 76

12

2y 32

81

1

y 3140

19

9ab23

0

1

x 56

256

9

8a34

3x2y

b3

a4

94

4

5.4 cm

3x2

4y 43

x 34

y3

x22x4

5y5

x 16 y

16 y

18

6.9 cm10.1 cm


35

Name Class Date

6-4 Practice Form KRational ExponentsSimplify each expression.

1. 16 14 2. (23)

13 ? (23)

13 ? (23)

13 3. 5

12 ? 45

12

!4 16

Write each expression in radical form.

4. x 14 5. x

45 6. x

29


7. !3 2 8. "3 2x2 9. "3 (2x)2

10. Bone loss for astronauts may be prevented with an apparatus that rotates

to simulate gravity. In the formula N 5 a0.5

2pr 0.5, N is the rate of rotation in

revolutions per second, a is the simulated acceleration in m/s2, and r is the radius of the apparatus in meters. How fast would an apparatus with the following radii have to rotate to simulate the acceleration of 9.8 m/s2 that is due to Earths gravity?

a. r 5 1.7 m b. r 5 3.6 m c. r 5 5.2 m d. Reasoning Would an apparatus with radius 0.8 m need to spin faster or

slower than the one in part (a)?

2

4!x

2 13

23

0.382 rev/s0.263 rev/s0.218 rev/s

faster

5"x 4

A2x 2B 13

15

9"x 2

(2x) 23


36

Name Class Date

6-4 Practice (continued) Form KRational ExponentsSimplify each number.

11. (2216) 13 12. 2431.2 13. 3220.4

!32216

Find each product or quotient. To start, rewrite the expression using exponents.

14. A!4 6B A!3 6B 15. "5 x210%x2 16. !20 ? !

3 135

5 Q6 14RQ6 13R

Simplify each number.

17. (125) 23 18. (216)

23(216)

23 19. (2243)

25

Write each expression in simplest form. Assume that all variables are positive.

20. Q16x28R234 21. Q8x15R2 13 22. a x2x210

b13

23. Error Analysis Explain why the following simplifi cation is incorrect. What is the correct simplifi cation?

5Q4 2 5 12R5 5(4) 2 5Q5 12R 5 20 2 25 12 5 15

26

12"6 7

25

x 68

You cannot multiply 5 and 5

12 together by multiplying bases. You

have to rewrite 5 as 51 and combine the exponents; 20 2 5!5.

729

5!x

1296

12x 5

14

6 6"5 5

9

x 4


37

Name Class Date

6-4 Standardized Test Prep Rational ExponentsMultiple Choice


1. What is 1213 ? 45

13 ? 50

13 in simplest form?

!27,000 30 10713 27,000 2. What is x

13 ? y


x3"y3 "xy3 "3 (xy)2 "3 xy2 3. What is x

13 ? x

12 ? x


x 1312 x

124 x

19 x

524

4. What is x23y13x

12y

34

6in simplest form? xy

52 x 7y

52

1

xy 52

x

y 52

5. What is (232x10 y35)215 in simplest form?

2x2y7 2 2

x2y7 2

1

2x2 y7

2

x2 y7

Short Response

6. Th e surface area S, in square units, of a sphere with volume V, in cubic units, is given by the formula S 5 p

13(6V )

23. What is the surface area of a sphere with

volume 43 mi3? Show your work.

[2] S 5 13(6V)

23 5

13 c6Q43R d

235

13 (8)

23 5 4

13 mi

2

[1] appropriate method but some computational errors[0] incorrect answer and no work shown OR no answer given

B

I

A

I

C


38

Name Class Date

6-4 Enrichment Rational ExponentsPower Games

Each problem below involves rational exponents. Some of the problems are tricky. Good luck!

1. Begin with any positive number. Call it x. Divide x by 2. Call the result r. Now follow these directions carefully. You may use a calculator.

a. Divide x by r. Call the result q. b. Add q and r. Call the result s. c. Divide s by 2. Call the result r. d. Go back to step a.

Repeat steps a d until r no longer changes. What is the relationship between the original x and the fi nal result?

2. If we take the square root of a number 6 times, it would look like this:

'&$#"!x Rewrite the expression above using rational exponents.

Simplify the expression above. Express the denominator of the exponent as a power of 2.

If you were to take the square root of a number 10 times, what would the denominator of the exponent be equal to if you use rational exponents? 12 times?

Choose any number and repeatedly take the square root. What number is the answer approaching?

Does the answer appear to approach the same number if you change the number you choose?

In Exercises 36, assume that the square roots and the operations inside them repeat forever.

3. How much is $2 3 #2 3 "2 3 !2 3c? (Hint: Let y 5 $2 3 #2 3 "2 3 !2 3c . Th en use substitution and solve the equation y 5 "2 3 y.)

4. How much is $2 1 #2 1 "2 1 !2 1c? 5. How much is $2 2 #2 2 "2 2 !2 2c? 6. How much is $2 4 #2 4 "2 4 !2 4c?

nal r 5 !x

aaaaax12b12b12b12b12b12x

164; 26

210; 212

1

yes

2

2

1

3"2


39

Name Class Date

6-4 Reteaching Rational ExponentsYou can simplify a number with a rational exponent by converting the expression to a radical expression:

x1n 5

n!x, for n . 0 912 5 2!9 5 3 813 5 !3 8 5 2You can simplify the product of numbers with rational exponents m and n by raising the number to the sum of the exponents using the rule

am ? an 5 am1n

Problem

What is the simplifi ed form of each expression?

a. 3614 ? 36

14

3614 ? 36

14 5 36

141

14 Use am ? an 5 am1n.

5 3612 Add.

5 2!36 Use x1n 5 n!x . 5 6 Simplify.

b. Write Q6x23R Q2x34R in simplifi ed form. Q6x23R Q2x34R 5 6 ? 2 ? x23 ? x34 Commutative and Associative

Properties of Multiplication

5 6 ? 2 ? x231

34 Use xm ? xn 5 xm1n.

5 12x1712 Simplify.

Exercises

Simplify each expression. Assume that all variables are positive.

1. 513 ? 5

23 2. Q2y14R Q3y13R 3. (211)13 ? (211)13 ? (211)13

4. 2y23 y

15 5. 5

14 ? 5

14 6. Q23x 16R Q7x 26R

5 6y 7

12 211

221!x!52y 1315


40

Name Class Date

6-4 Reteaching (continued)Rational ExponentsTo write an expression with rational exponents in simplest form, simplify all exponents and write every exponent as a positive number using the following rules for a 2 0 and rational numbers m and n:

a2n 51

an

1

a2m5 am (am)n 5 amn (ab)m 5 ambm

Problem

What is A8x9y23B223 in simplest form?(8x9y23)2

23 5 A23 x9 y23B223 Factor any numerical coef cients.

5 A23B223 Ax9B223 Ay23B223 Use the property (ab)m 5 ambm. 5 222x26y2 Multiply exponents, using the property (am)n 5 amn.

5y2

22x6 Write every exponent as a positive number.

5y2

4x6 Simplify.

Exercises

Write each expression in simplest form. Assume that all variables are positive.

7. A16x2 y8B212 8. Az23B19 9. Q2x14R4 10. A25x26 y2B12 11. A8a23 b9B23 12. a16z4

25x8b212

13. a x2y21b15 14. A27m9 n23B223 15. a32r2

2s4b14

16. A9z10B32 17. (2243)215 18. ax25y

12

b10

14xy4 16x

2r 12

s

5y

x34b6

a25x4

4z2

x4y52

13

n2

9m6

27z15

1

z 13

x25y

15


41

Name Class Date

6-5 Additional Vocabulary Support Solving Square Root and Other Radical Equations Problem

Solve the equation 4"3 (y 1 2)2 1 3 5 19. Justify your steps. Th en check your solution.

4(y 1 2)23 1 3 5 19 Rewrite the radical using a rational exponent.

4(y 1 2)23 5 16 Subtract 3 from each side.

(y 1 2)23 5 4 Divide each side by 4.

c(y 1 2)23 d 32 5 432 Raise each side to the 32 power. (y 1 2) 5 8 Simplify.

y 5 6 Solve for y.

Check 4 "3 (6 1 2)2 1 3 0 19 Substitute 6 for y. 4"3 82 1 3 0 19 Add. 4 ? 4 1 3 0 19 Simplify the radical.

19 5 19 Simplify.

Exercise

Solve the equation 9"(2x 2 4)4 1 2 5 38. Justify your steps. Th en check your solution.

9(2x 2 4)42 1 2 5 38

9(2x 2 4)2 5 36

(2x 2 4)2 5 4

c(2x 2 4)2 d 12 5 412 (2x 2 4) 5 2

x 5 3

Check 9"(2 ? 3 2 4)4 1 2 0 38 9"16 1 2 0 38

38 5 38

Rewrite the radical using a rational exponent.

Divide each side by 9.

Raise each side to the 12 power.

Simplify.

Solve for x.

Substitute 3 for x.

Simplify the expression under the radical sign.

Simplify.

Subtract 2 from each side and simplify the exponent.


42

Name Class Date

6-5 Think About a PlanSolving Square Root and Other Radical EquationsTraf c Signs A stop sign is a regular octagon, formed by cutting triangles off the corners of a square. If a stop sign measures 36 in. from top to bottom, what is the length of each side?

Understanding the Problem

1. How can you use the diagram at the right to fi nd a relationship between s and x?

.

2. How can you use the diagram at the right to fi nd another relationship between s and x?

.

3. What is the problem asking you to determine?

Planning the Solution

4. What are two equations that relate s and x?

5. How can you use your equations to fi nd s?

.

Getting an Answer

6. Solve your equations for s.

7. Is your answer reasonable? Explain.

.

x

x

x

s

s

36 in.

Since the triangles are right triangles, use the

Pythagorean Theorem to relate s and x

The length of a side of the square, which is s 1 2x, is the same as the height of the

stop sign from top to bottom

Yes; the length of one side of the stop sign is a little more than a third of the total

height of the sign

the length s of each side of the stop sign

Solve the rst equation for x and substitute the result into the second equation

2x2 5 s2; 2x 1 s 5 36

about 14.9 in.

Name Class Date


43

6-5 Practice Form GSolving Square Root and Other Radical EquationsSolve.

1. 5!x 1 2 5 12 2. 3!x 2 8 5 7 3. !4x 1 2 5 8 4. !2x 2 5 5 7 5. !3x 2 3 2 6 5 0 6. !5 2 2x 1 5 5 12 7. !3x 2 2 2 7 5 0 8. !4x 1 3 1 2 5 5 9. !33 2 3x 5 3 10. !3 2x 1 1 5 3 11. !3 13x 2 1 2 4 5 0 12. !3 2x 2 4 5 22Solve.

13. (x 2 2) 13 5 5 14. (2x 1 1)

13 5 23 15. 2x

34 5 16

16. 2x 13 2 2 5 0 17. x

12 2 5 5 0 18. 4x

32 2 5 5 103

19. (7x 2 3)12 5 5 20. 4x

12 2 5 5 27 21. x

16 2 2 5 0

22. (2x 1 1)13 5 1 23. (x 2 2)

23 2 4 5 5 24. 3x

43 1 5 5 53

25. Th e formula P 5 4"A relates the perimeter P, in units, of a square to its area A, in square units. What is the area of the square window shown below?

26. Th e formula A 5 6V 23 relates the surface area A, in square units, of a cube to

the volume V, in cubic units. What is the volume of a cube with surface area 486 in.2?

27. A mound of sand at a rock-crushing plant is growing over time. Th e equation t 5 !3 5V 2 1 gives the time t, in hours, at which the mound has volume V, in cubic meters. When is the volume equal to 549 m3?

Perimeter: 24 ft

4

27

17

13

127

1

4

0

36 ft2

729 in.3

14 h

29, 225 8, 28

64 64

25 9

214 16

5 22

32 8

13 222

25 9

Name Class Date


44

28. City offi cials conclude they should budget s million dollars for a new library building if the population increases by p thousand people in a ten-year census. Th e formula s 5 2 1 13(p 1 1)

25 expresses the relationship between population and

library budget for the city. How much can the population increase without the city going over budget if they have $5 million for a new library building?

Solve. Check for extraneous solutions.

29. !x 1 1 5 x 2 1 30. !2x 1 1 5 23 31. (x 1 7)

12 5 x 2 5 32. (2x 2 4)

12 5 x 2 2

33. !x 1 2 5 x 2 18 34. !x 1 6 5 x 35. (2x 1 1)

12 5 25 36. (x 1 2)

12 5 10 2 x

37. !x 1 1 5 x 1 1 38. !9 2 3x 5 3 2 x 39. !3 2x 2 4 5 22 40. 2!5 5x 1 2 2 1 5 3 41. !4x 1 2 5 !3x 1 4 42. !7x 2 6 2 !5x 1 2 5 0 43. 2(x 2 1)

12 5 (26 1 x)

12 44. (x 2 1)

12 2 (2x 1 1)

14 5 0

45. !2x 2 !x 1 1 5 1 46. !7x 2 1 5 !5x 1 5 47. (7 2 x)

12 5 (2x 1 13)

12 48. (x 2 7)

12 5 (x 1 5)

14

49. !x 1 9 2 !x 5 1 50. !3 8x 2 !3 6x 2 2 5 0 51. A clothing manufacturer uses the model a 5 !f 1 4 2 !36 2 f to estimate

the amount of fabric to order from a mill. In the formula, a is the number of apparel items (in hundreds) and f is the number of units of fabric needed. If 400 apparel items will be manufactured, how many units of fabric should be ordered?

52. What are the lengths of the sides of the trapezoid shown at the right if the perimeter of the trapezoid is 17 cm?

6-5 Practice (continued) Form GSolving Square Root and Other Radical Equations

x 1

x

2V x2V x

242,000

3

9

23

no solution

21, 0

22

2

10

8

22

16

32

x 5 4 cm, 2!x 5 4 cm, x 1 1 5 5 cm

21

11

3

4

4

6

0, 3

7

no solution

2, 4

9

Name Class Date


45

Solve. To start, rewrite the equation to isolate the radical.

1. !x 1 2 2 2 5 0 2. !2x 1 3 2 7 5 0 3. 2 1 !3x 2 2 5 6!x 1 2 5 2

Solve.

4. 2(x 2 2) 23 5 50 5. 2(x 1 3)

32 5 54 6. (6x 2 5)

13 1 3 5 22

7. Th e formula d 5 2# Vph relates the diameter d, in units, of a cylinder to its volume V, in cubic units, and its height h, in units. A cylindrical can has a diameter of 3 in. and a height of 4 in. What is the volume of the can to the nearest cubic inch?

8. Writing Explain the diff erence between a radical equation and a polynomial equation.

9. Reasoning If you are solving 4(x 1 3) 34 5 7, do you need to use the absolute

value to solve for x? Why or why not?

6-5 Practice Form KSolving Square Root and Other Radical Equations

223

127 and 2123 6 220

28 in.3

6

A radical equation has a variable in a radicand or a variable with arational exponent, while a polynomial equation has a variable with whole number exponents.

No; the numerator of the exponent 34 is not even.

Name Class Date


46

6-5 Practice (continued) Form KSolving Square Root and Other Radical EquationsSolve. Check for extraneous solutions. First, isolate a radical, then square each side of the equation.

10. !4x 1 5 5 x 1 2 11. !23x 2 5 2 3 5 x 12. !x 1 7 1 5 5 xA!4x 1 5B2 5 (x 1 2)2

13. !2x 2 7 5 !x 1 2 14. !3x 1 2 2 !2x 1 7 5 0 15. !2x 1 4 2 2 5 !xA!2x 2 7B2 5 A!x 1 2B2

16. Find the solutions of !x 1 2 5 x . a. Are there any extraneous solutions? b. Reasoning How do you know the answer to part (a)?

17. A fl oor is made up of hexagon-shaped tiles. Each hexagon tile has an area of 1497 cm2. What is the length of each side of the hexagon? (Hint: Six equilateral triangles make one hexagon.)

s

s!32

1 and 21

22 9

9

5 0 and 16

221

Substitute the solutionsinto the original equation. If a solution does not make the equation true, then the solution is extraneous.

about 24 cm


47

Name Class Date

Gridded Response

Solve each exercise and enter your answer in the grid provided.

1. What is the solution? !2x 2 4 2 3 5 1

2. What is the solution? 5x 12 2 8 5 7

3. What is the solution? !2x 2 6 5 3 2 x

4. What is the solution? !5x 2 3 5 !2x 1 3

5. Keplers Th ird Law of Orbital Motion states that the period P (in Earth years) it takes a planet to complete one orbit of the sun is a function of the distance d (in astronomical units, AU) from the planet to the sun. Th is relationship is P 5 d

32. If it takes Neptune 165 years to orbit the sun, what is the distance

(in AU) of Neptune from the sun? Round your answer to two decimal places.

6-5 Standardized Test Prep Solving Square Root and Other Radical Equations

1. 2. 3. 4. 5.

Answers

9

765

3210

9876543210

987654

210

9876543210

9876543210

9876543210

3

8

43

89

765

32

0

99876543210

987654

210

9876543210

9876543210

9876543210

3

8

43

8

11

9

765

3210

9876543210

987654

210

9876543210

9876543210

9876543210

3

8

43

89

765

3210

9876543210

987654

210

9876543210

9876543210

9876543210

3

8

43

89

76543210

9876543210

987654

210

9876543210

9876543210

9876543210

3

8

1 0

10

9

9

3

3

22

2

3 0 . 0 8

3

0 0

8


48

Name Class Date

When solving radical equations you will often get an extraneous solution. You can use a graph to explain why an algebraic answer is not a solution.

1. Solve the equation !x 1 2 5 x 2 4. Is there an extraneous solution?

2. To analyze this equation with a graph, rewrite the equation as a system of two equations. What two equations can you write?

3. Graph the two equations.

4. Explain how you fi nd the solution to this system of equations on your graph. What is the solution?

5. How can you use the solution from the graph of the system of equations to help you solve the original equation !x 1 2 5 x 2 4?

6. How can you tell from your graph that one of your algebraic answers is an extraneous solution?

Solve each equation. Graph each equation as a system to determine if there are any extraneous solutions.

7. !4x 1 1 5 3 8. x 5 !6 2 x 9. !x 1 1 5 x 2 1

6-5 Enrichment Solving Square Root and Other Radical Equations

7; 2 is an extraneous solution.

y 5 !x 1 2 and y 5 x 2 4

Answers may vary. Sample: On a graph the solution to a system of equations is the point of intersection; the solution for this system is (7, 3).

Answers may vary. Sample: The x-coordinate of the solution to the system is the solution to the original equation.

Answers may vary. Sample: Because there is only one point of intersection, there can only be one solution to the equation.

2; no extraneous solutions



4 8

8

4

4

8

x

y

O8 4

4 8

8

4

4

8 y

xO8 4 4 84

8

4

4

8

x

y

4 4 8

8

4

4

8

x

y


49

Name Class Date

6-5 Reteaching Solving Square Root and Other Radical EquationsEquations containing radicals can be solved by isolating the radical on one side of the equation, and then raising both sides to the same power that would undo the radical.

Problem

What is the solution of the radical equation? 2!2x 1 2 2 2 5 10 2!2x 1 2 2 2 5 10 2!2x 1 2 5 12 Add 2 to each side. !2x 1 2 5 6 Divide each side by 2. (!2x 1 2)2 5 62 Square each side to undo the radical. 2x 1 2 5 36 Simplify.

2x 5 34 Subtract 2 from each side.

x 5 17 Divide each side by 2.

Check the solution in the original equation.

Check

2!2x 1 2 2 2 5 10 Write the original equation. 2!2(17) 1 2 2 2 0 10 Replace x by 17. 2!36 2 2 0 10 Simplify. 12 2 2 0 10

10 5 10

Th e solution is 17.

Exercises

Solve. Check your solutions.

1. x 12 5 13 2. 3!2x 5 12 3. !3x 1 5 5 11

4. (3x 1 4)12 2 1 5 4 5. (6 2 x)

12 1 2 5 5 6. !3x 1 13 5 4

7. (x 1 2) 12 2 5 5 0 8. !3 2 2x 2 2 5 3 9. !3 5x 1 2 2 3 5 0

169

7 23 1

211 523

8 12


50

Name Class Date

6-5 Reteaching (continued) Solving Square Root and Other Radical EquationsAn extraneous solution may satisfy equations in your work, but it does not make the original equation true. Always check possible solutions in the original equation.

Problem

What is the solution? Check your results. !17 2 x 2 3 5 x !17 2 x 2 3 5 x !17 2 x 5 x 1 3 Add 3 to each side to get the radical alone on one side of the equal sign.

A!17 2 xB2 5 (x 1 3)2 Square each side. 17 2 x 5 x2 1 6x 1 9

0 5 x2 1 7x 2 8 Rewrite in standard form.

0 5 (x 2 1)(x 1 8) Factor.

x 2 1 5 0orx 1 8 5 0 Set each factor equal to 0 using the Zero Product Property.

x 5 1 or x 5 28

Check

!17 2 x 2 3 5 x !17 2 x 2 3 5 x !17 2 1 2 3 0 1 !17 2 (28) 2 3 0 28 !16 2 3 0 1 !25 2 3 0 28 1 5 1 2 2 28

Th e only solution is 1.

Exercises

Solve. Check for extraneous solutions.

10. !5x 1 1 5 !4x 1 3 11. !x2 1 3 5 x 1 1 12. !3x 5 !x 1 6 13. x 5 !x 1 7 1 5 14. x 2 3!x 2 4 5 0 15. !x 1 2 5 x 2 4 16. !2x 2 10 5 x 2 5 17. !3x 2 6 5 2 2 x 18. !x 2 1 1 7 5 x 19. !5x 1 1 5 !3x 1 15 20. !x 1 9 5 x 1 7 21. x 2 !x 1 2 5 40

5, 7 2

2

9 16 7

3solutionno

7 25

10

47


51

Name Class Date

6-6 Additional Vocabulary Support Function OperationsDarnell wrote the steps to compose the following functions on index cards, but the cards got mixed up.

Let f (x) 5 x 1 7 and g(x) 5 x3. What is (g + f )(24)?

Use the note cards to write the steps in order.

1. First,

.

2. Second,

.

3. Then,

.

4. Finally,

.

Subtract 4 from 7.

Raise 3 to the 3rd power.

Substitute 24 for x in f(x).

Substitute 3 into g(x).

substitute 24 for x in f(x)

subtract 4 from 7

substitute 3 into g(x)

raise 3 to the 3rd power


52

Name Class Date

6-6 Think About a Plan Function OperationsSales A salesperson earns a 3% bonus on weekly sales over $5000. Consider the following functions.

g(x) 5 0.03x h(x) 5 x 2 5000

a. Explain what each function above represents.

b. Which composition, (h + g)(x) or (g + h)(x), represents the weekly bonus? Explain.

1. What does x represent in the function g(x)?

2. What does the function g(x) represent?

3. What does x represent in the function h(x)?

4. What does the function h(x) represent?

5. What is the meaning of (h + g)(x)?

.

6. Assume that x is $7000. What is (h + g)(x)?

7. What is the meaning of (g + h)(x)?

.

8. Assume that x is $7000. What is (g + h)(x)?

9. Which composition represents the weekly bonus? Explain

.

the sales amount used to calculate a 3% bonus

the bonus earned by the salesperson on sales

the total weekly sales made by the salesperson

the weekly sales over $5000 made by the salesperson

First multiply the value of x by 0.03, then subtract 5000 from the result

(g + h)(x) represents the weekly bonus because you must rst nd the sales amount

over 5000 by subtracting 5000 from the weekly sales, and then you multiply the result

by the bonus percent as a decimal, or 0.03

First subtract 5000 from the value of x, then multiply the result by 0.03

4790

60

Name Class Date


53

6-6 Practice Form GFunction OperationsLet f (x) 5 4x 21 and g(x) 5 2x2 1 3. Perform each function operation and then fi nd the domain.

1. ( f 1 g)(x) 2. ( f 2 g)(x) 3. (g 2 f )(x)

4. ( f ? g) (x) 5. fg (x) 6.

gf (x)

Let f (x) 5 2x and g(x) 5 !x 21. Perform each function operation and then fi nd the domain of the result.

7. ( f 1 g)(x) 8. ( f 2 g)(x) 9. (g 2 f )(x)

10. ( f ? g)(x) 11. fg (x) 12.

gf (x)

Let (x) 5 23x 1 2, g(x) 5 x5 , h(x) 5 22x2 1 9, and j(x) 5 5 2 x. Find each value or expression.

13. ( f + j)(3) 14. ( j + h)(21) 15. (h + g)(25)

16. (g + f )(a) 17. (x) 1 j(x) 18. (x) 2 h(x)

19. (g + f )(25) 20. ( f + g)(22) 21. 3(x) 1 5g(x)

22. g( f (2)) 23. g( f (x)) 24. f (g(1))

25. A video game store adds a 25% markup on each of the games that it sells. In addition to the manufacturers cost, the store also pays a $1.50 shipping charge on each game.

a. Write a function to represent the price f (x) per video game after the stores markup.

b. Write a function g(x) to represent the manufacturers cost plus the shipping charge.

c. Suppose the manufacturers cost for a video game is $13. Use a composite function to fi nd the cost at the store if the markup is applied after the shipping charge is added.

d. Suppose the manufacturers cost for a video game is $13. Use a composite function to fi nd the cost at the store if the markup is applied before the shipping charge is added.

2x2 1 4x 1 2; all real numbers

8x3 2 2x2 1 12x 23; all real numbers

2x 1 "x 2 1; x L 0

2x"x 2 2x; x L 0 2x!x 2 1; x L 0 and x u 1

2x 2 "x 1 1; x L 0 22x 1 "x 2 1; x L 0"x 2 1

2x ; x S 0

22x2 + 4x 24; all real numbers

4x 2 12x2 1 3

; all real numbers 2x2 1 3

4x 2 1 ; all real numbers except 14

2x2 2 4x 1 4; all real numbers

24

175

2 45 23x 1 2

5

165

23a 1 25

22

24x 1 7 2x2 2 3x 2 7

28x 1 6

75

7

f(x) 5 1.25x

g(x) 5 x 1 1.5

f(g(13)) N $18.13

g(f(13)) 5 $17.75

Name Class Date


54

26. Th e formula V 5 43 r 3 expresses the relationship between the volume V and

radius r of a sphere. A weather balloon is being infl ated so that the radius is changing with respect to time according to the equation r 5 t 1 1, where t is the time, in minutes, and r is the radius, in feet.

a. Write a composite function f (t) to represent the volume of the weather balloon after t minutes. Do not expand the expression.

b. Find the volume of the balloon after 5 minutes. Round the answer to two decimal places. Use 3.14 for .

27. A boutique prices merchandise by adding 80% to its cost. It later decreases by 25% the price of items that do not sell quickly.

a. Write a function f (x) to represent the price after the 80% markup. b. Write a function g(x) to represent the price after the 25% markdown. c. Use a composition function to fi nd the price of an item, after both price

adjustments, that originally costs the boutique $150. d. Does the order in which the adjustments are applied make a diff erence?

Explain.

28. A department store has marked down its merchandise by 25%. It later decreases by $5 the price of items that have not sold.

a. Write a function f (x) to represent the price after the 25% markdown. b. Write a function g(x) to represent the price after the $5 markdown. c. Use a composition function to fi nd the price of a $50 item after both price

adjustments. d. Does the order in which the adjustments are applied make a diff erence?

Explain.

Let g(x) 5 x2 2 5 and h(x) 5 3x 1 2. Perform each function operation.

29. (h + g)(x) 30. g(x) ? h(x) 31. 22g(x) 1 h(x)

6-6 Practice (continued) Form GFunction Operations

f(t) 5 43 (t 1 1)3

904.32 ft3

f(x) 5 1.8x

f(x) 5 0.75xg(x) 5 x 2 5

g(x) 5 0.75x

g(f (150)) 5 $202.50

No; it doesnt matter whether you rst multiply by 0.75 or by 1.8.

Yes; multiplying by 0.75 and then subtracting by 5 is different than subtracting by 5 and then multiplying by 0.75.

g(f (50)) 5 $32.50

3x2 2 13 3x3 1 2x2 2 15x 2 10 22x2 1 3x 1 12

Name Class Date


55

6-6 Practice Form KFunction OperationsLet f (x) 5 4x 1 8 and g(x) 5 2x 2 12. Perform each function operation and then fi nd the domain of the result.

1. ( f 1 g)(x) 2. ( f 2 g)(x) 3. ( f ? g)(x) 4. a fgb(x)f (x) 1 g(x)

Let f (x) 5 x 1 2 and g(x) 5 !x 2 1. Perform each function operation and then fi nd the domain of the result.

5. ( f 1 g)(x) 6. ( f ? g)(x) 7. a fgb(x) 8. agf b(x)

Let f (x) 5 x 2 2 and g(x) 5 x2. Find each value. To start, use the defi nition of composing functions to fi nd a function rule.

9. (gf )(4) 10. ( fg)(21) 11. (gf )(23)f (4) 5 4 2 2 5 2

Let f (x) 5 !x and g(x) 5 (x 1 2)2. Find each value. 12. ( fg)(25) 13. ( fg)(0) 14. (gf )(4)

(f 1 g) (x) 5 6x 2 4; all real numbers

(f 1 g)(x)5 x 1 !x 1 1;all x L 0

(f ? g)(x)5 x!x 2 x 1 2!x 2 2;all x L 0

Q fgR(x) 5 x 1 2!x 2 1; all x L 0, x u 1

Qgf R(x) 5 !x 2 1x 1 2 ; all x L 0

(f 2 g) (x)

5 2x 1 20; all real numbers

(f ? g) (x)

5 8x2 2 32x 2 96; all real numbers

Q fgR(x) 5 2x 1 4x 2 6 ; all real numbers, x u 6

421 25

3 2 16

Name Class Date


56

6-6 Practice (continued) Form KFunction Operations 15. A car dealer off ers a 15% discount off the list price x of any car on the lot. At the same

time, the manufacturer off ers a $1000 rebate for each purchase of a car. a. Write a function f (x) to represent the price after discount. b. Write a function g (x) to represent the price after the $1000 rebate. c. Suppose the list price of a car is $18,000. Use a composite function to fi nd the price

of the car if the discount is applied before the rebate. d. Suppose the list price of a car is $18,000. Use a composite function to fi nd the price

of the car if the discount is applied after the rebate. e. Reasoning Between parts (c) and (d), will the dealer want to apply the

discount before or after the rebate? Why?

16. Error Analysis f (x) 5 2!x and g(x) 5 3x 2 6. Your friend gives a domain for a fgb(x) as x $ 0. Is this correct? If not, what is the correct domain?

Let f (x) 5 2x2 2 3 and g(x) 5 x 1 12 . Find each value.

17. f (g(2)) 18. g( f (23)) 19. ( ff )(21)

20. Reasoning A local bookstore has a sale on all their paperbacks giving a 10% discount. You also received a coupon in the mail for $4 off your purchase. If you buy 2 paperbacks at $8 each, is it less expensive for you to apply the discount before the coupon or after the coupon? How much will you save? before the coupon; $.40

f(x) 5 0.85xg(x) 5 x 2 1000

$14,300

$14,450

After; they will make more money selling the car for a higher price.

No; the correct domain is x L 0, x u 2.

32

8 21


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Name Class Date

Multiple Choice


1. Let f (x) 5 22x 1 5 and g(x) 5 x3. What is (g 2 f )(x)?

x3 2 2x 1 5 2x3 2 2x 1 5

x3 1 2x 2 5 2x3 1 2x 2 5

2. Let f (x) 5 3x and g(x) 5 x2 1 1. What is ( f g)(x)?

9x2 1 3x 9x2 1 1 3x3 1 3x 3x3 1 1

3. Let f (x) 5 x2 2 2x 2 15 and g(x) 5 x 1 3. What is the domain of fg (x)?

all real numbers x 2 23

x 2 5, 23 x . 0

4. Let f (x) 5 !x 1 1 and g(x) 5 2x 1 1. What is (g + f )(x)? 2!x 1 3 !2x 1 1 1 1 2x!x 1 2x 1 !x 1 1 2x 1 !x 1 2

5. Let f (x) 5 1x and g(x) 5 x2 2 2. What is ( f + g)(23)?

179 17 2

179 2

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Short Response

6. Suppose the function f (x) 5 0.035x represents the number of U.S. dollars equivalent to x Russian rubles and the function g(x) 5 90x represents the number of Japanese yen equivalent to x U.S. dollars. Write a composite function that represents the number of Japanese yen equivalent to x Russian rubles. Show your work.

6-6 Standardized Test PrepFunction Operations

[2] (g f )(x) 5 g(f(x)) 5 90(0.035x) 5 3.15x

B

H

C

F

B

[1] appropriate method but with one computational error

[0] incorrect answer and no work shown OR no answer given


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Name Class Date

Composition and Linear FunctionsTwo functions f (x) and g(x) are equal if they have the same domains and the same value for each point in their domain. Suppose that f (x) 5 Ax 1 B and g(x) 5 Cx 1 D are two linear functions both of whose domains are the set of real numbers.

1. If f (x) 5 g(x), what can you conclude by examining the values of f and g at x 5 0?

2. Use your conclusion to eliminate D from the defi nition of g(x).

3. What equation results from ex

ch6 ans

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