Chapter 3

Exercise Set 3.1

Simplify the expression, writing it without any negative exponents.

1.

(−2x4 )3

(-2x

4

)

3

2.

(6y)3

(6y)

3

3.

(−3a4bc5 )2

(-3a

4

bc

5

)

2

4.

−3(−2x2 )3

-3(-2x

2

)

3

5.

(−7a4 )2

(-7a

4

)

2

6.

a2 (a4b3c)5

a

2

(a

4

b

3

c)

5

7.

(xy2z3)4

(x3y2z)3

(xy

2

z

3

)

4

(x

3

y

2

z)

3

8.

(2y3)4

2y5

(2y

3

)

4

2y

5

9.

(2a3)2 (3a4 )(a3)4

(2a

3

)

2

(3a

4

)

(a

3

)

4

10.

(−x3y)2 z4( )3

(-x

3

y)

2

z

4

()

3

11.

−u2v3

4u4v⎛

⎝⎜

⎞

⎠⎟

2

-u

2

v

3

4u

4

v

æ

è

ç

ö

ø

÷

2

12.

2xy6zxy2z3

⎛

⎝⎜

⎞

⎠⎟

3

2xy

6

z

xy

2

z

3

æ

è

ç

ö

ø

÷

3

13.

(2x3)−3

(2x

3

)

-3

14.

(ab2 )−7

(ab

2

)

-7

15.

u3u−9

u

3

u

-9

16.

(−2x5 )−2

(-2x

5

)

-2

17.

(x2y−3)−4

(x

2

y

-3

)

-4

18.

(−u−4 )3(2u5 )−2

(-u

-4

)

3

(2u

5

)

-2

19.

(−4x2y−7 )3

(-4x

2

y

-7

)

3

20.

−3x−2y( )2(2x3y−4 )−2

-3x

-2

y

()

2

(2x

3

y

-4

)

-2

21.

(−2a3b−4c−7 )3

(-2a

3

b

-4

c

-7

)

3

22.

a−3b4

a−5b5

a

-3

b

4

a

-5

b

5

23.

a−1

5b4⎛

⎝⎜

⎞

⎠⎟

2

a

-1

5b

4

æ

è

ç

ö

ø

÷

2

24.

−2a4

b2⎛

⎝⎜

⎞

⎠⎟

−3

-2a

4

b

2

æ

è

ç

ö

ø

÷

-3

25.

9yy−5⎛

⎝⎜

⎞

⎠⎟

−1

9y

y

-5

æ

è

ç

ö

ø

÷

-1

26.

x6

4x2⎛

⎝⎜

⎞

⎠⎟

−2

x

6

4x

2

æ

è

ç

ö

ø

÷

-2

27.

x8

x−2⎛

⎝⎜

⎞

⎠⎟

3x−3

2x⎛

⎝⎜

⎞

⎠⎟

2

x

8

x

-2

æ

è

ç

ö

ø

÷

3

x

-3

2x

æ

è

ç

ö

ø

÷

2

28.

(−a2b3)−2

a−4b2

(-a

2

b

3

)

-2

a

-4

b

2

29.

x−1yz−2

x−8y−5z⎛

⎝⎜

⎞

⎠⎟

−1

x

-1

yz

-2

x

-8

y

-5

z

æ

è

ç

ö

ø

÷

-1

30.

xy−2z−3

x2y3z−4⎛

⎝⎜

⎞

⎠⎟

−3

xy

-2

z

-3

x

2

y

3

z

-4

æ

è

ç

ö

ø

÷

-3

31.

(3ab2c) 2a2bc3

⎛

⎝⎜

⎞

⎠⎟

−2

(3ab

2

c)

2a

2

b

c

3

æ

è

ç

ö

ø

÷

-2

32.

14x2y−5 3 x

−4

y−2⎛

⎝⎜

⎞

⎠⎟

2

1

4

x

2

y

-5

3

x

-4

y

-2

æ

è

ç

ö

ø

÷

2

33.

x5

2y−3⎛

⎝⎜

⎞

⎠⎟

36x−8

y⎛

⎝⎜

⎞

⎠⎟

2

x

5

2y

-3

æ

è

ç

ö

ø

÷

3

6x

-8

y

æ

è

ç

ö

ø

÷

2

Find the value of the algebraic expression at the specified values of its variable or variables without using a calculator. Simplify the expression before evaluating. Check your answer using a calculator.

34.

x3x−4; x = 5

x

3

x

-4

;x=5

35.

(x−2 )3; x = −2

(x

-2

)

3

;x=-2

36.

(2x−1)3; x = 3

(2x

-1

)

3

;x=3

37.

(x2 + y2 )−1; x = −1, y = 2

(x

2

+y

2

)

-1

;x=-1,y=2

38.

x−2 + y−2; x =1, y = 2

x

-2

+y

-2

;x=1,y=2

39.

(x + y)−1; x = 3, y = 5

(x+y)

-1

;x=3,y=5

40.

x−1 + y−1; x = 3, y = 5

x

-1

+y

-1

;x=3,y=5

41.

xy⎛

⎝⎜⎞

⎠⎟

−2

; x = −2, y = 3

x

y

æ

è

ç

ö

ø

÷

-2

;x=-2,y=3

42.

xy⎛

⎝⎜⎞

⎠⎟

−2

; x =1, y = 3

x

y

æ

è

ç

ö

ø

÷

-2

;x=1,y=3

43.

x2

x−3; x = −2

x

2

x

-3

;x=-2

44.

x−3

x−2; x = 7

x

-3

x

-2

;x=7

45.

x−2

y⎛

⎝⎜

⎞

⎠⎟

−1

; x = 3, y = 7

x

-2

y

æ

è

ç

ö

ø

÷

-1

;x=3,y=7

Exercise Set 3.2

Simplify the expression. Assume that all variables represent positive numbers.

1.

12x6

12x

6

2.

32x2y10

32x

2

y

10

3.

81x9

81x

9

4.

25x5y9

25x

5

y

9

5.

24z7

24z

7

6.

18y3z12

18y

3

z

12

7.

8x5y17

8x

5

y

17

8.

3x8y5z7

3x

8

y

5

z

7

9.

20x20

20

x

20

10.

x6

y10z2

x

6

y

10

z

2

11.

32x3

9y2

32x

3

9y

2

12.

y9z13

w12

y

9

z

13

w

12

13.

72x5y9

72x

5

y

9

14.

498x5

49

8x

5

15.

27y8

8z3

27y

8

8z

3

16.

52y8

45z12

52y

8

45z

12

17.

18x5

2x−2

18x

5

2x

-2

18.

x5yz8

x2y−5

x

5

yz

8

x

2

y

-5

19.

9x−2y6

45x−7y2

9x

-2

y

6

45x

-7

y

2

20.

33x2y6

9x−3

33x

2

y

6

9x

-3

Find the value of the algebraic expression at the specified values of its variable or variables in simplified form without a calculator. Simplify the expression before evaluating. Check your answer using a calculator.

21.

x3y4 ; x = 3, y = 2

x

3

y

4

;x=3,y=2

22.

x3y5 ; x = 4, y = 7

x

3

y

5

;x=4,y=7

23.

44x5 ; x = 7

44x

5

;x=7

24.

98x6 ; x = 2

98x

6

;x=2

25.

18x5

2x−2; x = 4

18x

5

2x

-2

;x=4

26.

x3y−4

x7y−2; x = 3, y = 5

x

3

y

-4

x

7

y

-2

;x=3,y=5

Exercise Set 3.3

Simplify the expression. Assume that all variables represent positive numbers.

1.

8x113

8x

11

3

2.

16x73

16x

7

3

3.

40x6y53

40x

6

y

5

3

4.

72x10y53

72x

10

y

5

3

5.

48y12

3

48

y

12

3

6.

64x25

y83

64x

25

y

8

3

7.

54x18

y273

54x

18

y

27

3

8.

32x28

27y153

32x

28

27y

15

3

Write each radical expression as an exponential expression and each exponential expression as a radical expression.

9.

13

1

3

10.

723

7

2

3

11.

53

5

3

12.

1x73

1

x

7

3

13.

x2 3

x

23

14.

7−5 2

7

-52

15.

x3 5

x

35

16.

x−5 3

x

-53

Evaluate each expression without a calculator. Check your answer using a calculator.

17.

827

3

8

27

3

18.

−643

-64

3

19.

325

32

5

20.

116

4

1

16

4

21.

−542

3

-

54

2

3

22.

−273

83

-27

3

8

3

23.

−35

965

-3

5

96

5

24.

53

403

5

3

40

3

25.

81 3

8

13

26.

641 4

64

14

27.

(−32)1 5

(-32)

15

28.

811 2

81

12

Find the value of the algebraic expression at the specified values of its variable or variables without using a calculator. Check your answer using a calculator.

29.

x2 3; x = 8

x

23

;x=8

30.

x3 4; x =16

x

34

;x=16

31.

(x3y)1 2; x = 2, y = 4

(x

3

y)

12

;x=2,y=4

32.

x4 3; x = 8

x

43

;x=8

33.

x4

y2⎛

⎝⎜

⎞

⎠⎟

3 2

; x = 2, y = 3

x

4

y

2

æ

è

ç

ö

ø

÷

32

;x=2,y=3

34.

(x2 − y)3 2; x = 5, y = −7

(x

2

-y)

32

;x=5,y=-7

35.

(x3 + y3)5 2; x =1, y = 2

(x

3

+y

3

)

52

;x=1,y=2

36.

(xy2 )−2 3; x = 2, y = −2

(xy

2

)

-23

;x=2,y=-2

37.

(x2 + y2 )−1 2; x = 3, y = 4

(x

2

+y

2

)

-12

;x=3,y=4

38.

x2

y2⎛

⎝⎜

⎞

⎠⎟

−5 2

; x = 3, y = 2

x

2

y

2

æ

è

ç

ö

ø

÷

-52

;x=3,y=2