Deck 8: Rational Exponents and Radicals

Question 1

Use the properties of exponents to simplify the expression. Write with positive exponents.
$4 x ^ { - 2 / 5 } ( x -^ { 11 / 5 } - 5 x )$

A) $4 x ^ { 9 / 5 } + 5 x ^ { - 2 / 5 }$
B) $\frac { 4 } { x ^ { 9 / 5 } } - \frac { 20 } { x ^ { - 2 / 5 } }$
C) $4 x ^ { 9 / 5 } - 20 x ^ { 3 / 5 }$
D) $4 x ^ { - 22 / 5 } - 20 x ^ { - 2 / 5 }$

Answer 1

C

Question 2

Write with positive exponents.
$x ^ { - 8 / 7 }$

A) $\frac { 1 } { x ^ { 7 / 8 } }$
B) $\frac { 1 } { x ^ { 8 / 7 } }$
C) $- x ^ { 8 / 7 }$
D) $\frac { - 1 } { x ^ { 8 / 7 } }$

Answer 2

B

Question 3

Use the properties of exponents to simplify the expression. Write with positive exponents.
$64 \left( \frac { a ^ { 3 } b ^ { - 6 } } { a ^ { - 3 } b ^ { 6 } } \right) ^ { 1 / 3 }$

A) $\frac { 16 a } { b ^ { 2 } }$
B) $\frac { 16 b ^ { 6 } } { a }$
C) $4 a ^ { 2 } b ^ { 6 }$
D) $\frac { 4 a ^ { 2 } } { b ^ { 4 } }$

Answer 3

D

Question 4

Simplify.
$\frac { y ^ { 3 / 4 } } { y ^ { 1 / 4 } }$

A) $y$
B) $y ^ { 3 / 4 }$
C) $y ^ { 1 / 2 }$
D) $\frac { 1 } { y }$

Question 5

Simplify.
$\frac { y ^ { 7 / 6 } } { y ^ { 5 / 6 } }$

A) $y ^ { 7 / 5 }$
B) $y ^ { 1 / 3 }$
C) $y ^ { 5 / 7 }$
D) $y ^ { 6 / 3 }$

Question 6

Simplify.
$$\left( x ^ { 4 / 3 } \right) ^ { 15 }$$

A) $x ^ { 4 / 45 }$
B) $x ^ { 20 }$
C) $x ^ { 49 / 5 }$
D) $x ^ { 4 / 5 }$

Question 7

Use the properties of exponents to simplify the expression. Write with positive exponents.
$\left. 4 a ^ { 1 / 7 } b ^ { 5 / 7 } \right) ^ { 3 }$

A) $4 a ^ { 3 / 7 } b ^ { 15 / 7 }$
В) $64 a ^ { 3 / 7 } b 15 / 7$
C) $64 ( \mathrm { ab } ) ^ { 15 / 7 }$
D) $12 \mathrm { a } ^ { 3 / 7 } \mathrm {~b} ^ { 5 / 7 }$

Question 8

Use the properties of exponents to simplify the expression. Write with positive exponents.
$\frac { x ^ { 3 / 2 } \cdot x ^ { - 4 / 7 } } { x ^ { - 2 / 3 } }$

A) $\frac { 1 } { x ^ { 11 / 42 } }$
B) $x ^ { 67 / 42 }$
C) $\mathrm { x } ^ { 11 / 42 }$
D) $\frac { 1 } { x ^ { 67 / 42 } }$

Question 9

Write with positive exponents.
$$2 ^ { - 1 / 4 } \cdot 5 ^ { 2 / 5 }$$

A) $10 ^ { 3 / 20 }$
B) $\frac { 1 } { 2 ^ { 1 / 4 \cdot 5 ^ { 2 / 5 } } }$
C) $\frac { 5 ^ { 2 / 5 } } { 2 ^ { 1 / 4 } }$
D) $- 10 ^ { 2 / 20 }$

Question 10

Evaluate the numerical expression.
$16^{1 / 2}$

A) 16
B) 2
C) 4
D) 8

Question 11

Simplify.
$$\left( \frac { 4 a ^ { - 4 } b ^ { 2 } } { b } \right) ^ { 2 }$$

A) $\frac { 16 b ^ { 3 } } { a ^ { 8 } }$
B) $\frac { 16 b ^ { 2 } } { a ^ { 8 } }$
C) $\frac { 16 b ^ { 6 } } { a ^ { 8 } }$
D) $\frac { 16 b ^ { 4 } } { a ^ { 8 } b ^ { 2 } }$

Question 12

Simplify.
$$y ^ { - 1 / 7 } \cdot y ^ { 5 / 14 }$$

A) $\mathrm { y } ^ { - 5 / 98 }$
B) $y ^ { 2 / 21 }$
C) $y ^ { 3 / 14 }$
D) $y ^ { 2 / 14 }$

Question 13

Write with positive exponents.
$$6 a ^ { - 3 / 8 } b ^ { 5 / 4 }$$

A) $\frac { 6 b ^ { 5 / 4 } } { a ^ { 3 / 8 } }$
B) $\frac { 6 } { a b ^ { 15 / 2 } }$
C) $\frac { b ^ { 5 / 4 } } { 6 a ^ { 3 / 8 } }$
D) $\frac { 1 } { 6 a ^ { 3 / 8 } b ^ { 5 / 4 } }$

Question 14

Write with positive exponents.
$5 ^ { - 1 / 4 }$

A) $\frac { 19 } { 4 }$
B) $- 5 ^ { 1 / 4 }$
C) $- \frac { 5 } { 4 }$
D) $\frac { 1 } { 51 / 4 }$

Question 15

Use the properties of exponents to simplify the expression. Write with positive exponents.
$$\left( x ^ { 1 / 3 } y ^ { 3 / 5 } \right) \left( x ^ { - 1 / 3 } y ^ { 3 / 2 } \right)$$

A) $\frac { y ^ { 9 / 10 } } { x ^ { 1 / 9 } }$
B) $\frac { y ^ { 9 / 10 } } { x ^ { 2 / 3 } }$
C) $x y ^ { 21 / 10 }$
D) $y ^ { 21 / 10 }$

Question 16

Use the properties of exponents to simplify the expression. Write with positive exponents.
$$\left( 2 x ^ { - 1 / 3 } y ^ { 1 / 3 } \right) \left( - 8 x ^ { 1 / 4 } y ^ { - 1 / 6 } \right)$$

A) $\frac { - 16 } { x ^ { 1 / 12 } y ^ { 1 / 18 } }$
B) $\frac { 16 x ^ { 1 / 12 } } { y ^ { 1 / 6 } }$
C) $\frac { - 16 y ^ { 1 / 6 } } { x ^ { 1 / 12 } }$
D) $\frac { - 6 } { x ^ { 1 / 12 } y ^ { 1 / 18 } }$

Question 17

Simplify.
$\left( \mathrm { b } ^ { 7 } \right) ^ { 2 / 7 }$

A) $b ^ { 2 }$
B) $b ^ { 2 / 49 }$
C) $b ^ { 1 / 2 }$
D) $b ^ { 9 / 7 }$

Question 18

Evaluate the numerical expression.
$81 ^ { 1 / 4 }$

A) 12
B) 36
C) 3
D) 243

Question 19

Use the properties of exponents to simplify the expression. Write with positive exponents.
$\frac { \left( - 5 x ^ { 5 / 3 } \right) ^ { 3 } } { x ^ { 3 / 5 } }$

A) $- 125 x ^ { 22 / 5 }$
B) $- 5 x ^ { 22 / 5 }$
C) $- 125 x ^ { 28 / 5 }$
D) $- 5 x ^ { 28 / 5 }$

Question 20

Use the properties of exponents to simplify the expression. Write with positive exponents.
$y ^ { 5 / 9 } \left( y ^ { 2 / 9 } - 5 y ^ { 3 / 9 } \right)$

A) $y ^ { 7 / 9 } - 5 y ^ { 8 / 9 }$
B) $\mathrm { y } ^ { 10 / 81 } - 5 \mathrm { y } ^ { 15 / 81 }$
C) $y ^ { 5 / 2 } - 5 y ^ { 5 / 3 }$
D) $y ^ { 3 / 9 } - 5 y ^ { 2 / 9 }$

Question 21

Evaluate if possible.
$\sqrt { \frac { 1 } { 81 } }$

A) 81
B) $\frac { 1 } { 6561 }$
C) $\frac { 1 } { 9 }$
D) 9

Question 22

Evaluate if possible.
$\sqrt { 81 }$

A) 9
B) 40
C) not a real number
D) 10

Question 23

Evaluate the numerical expression.
$- 32 ^ { 1 / 5 }$

A) $- 8$
B) $- 2$
C) 16
D) 32

Question 24

Find the requested function value for the given function. Round the answer to the nearest tenth when necessary.
If $f ( x ) = \sqrt { 2 x + 1 }$ , find the value of $f ( 5 )$ .

A) 10
B) 11
C) $4.5$
D) $3.3$

Question 25

Write the expression as one fraction with positive exponents.
$$3 ^ { - 1 / 5 } + x ^ { - 1 / 2 }$$

A) $\frac { x ^ { 1 / 2 } 3 ^ { 4 / 5 } } { 3 ^ { 1 / 5 } + x ^ { 1 / 2 } }$
B) $\frac { x ^ { 1 / 2 } 3 ^ { 1 / 5 } } { 3 ^ { 1 / 5 } + x ^ { 1 / 2 } }$
C) $\frac { \times ^ { 1 / 2 } + 3 ^ { 4 / 5 } } { 3 ^ { 1 / 5 } \times 1 / 2 }$
D) $\frac { \times ^ { 1 / 2 } + 3 ^ { 1 / 5 } } { 3 ^ { 1 / 5 } x ^ { 1 / 2 } }$

Question 26

Write the expression as one fraction with positive exponents.
$$3 y ^ { 5 / 4 } - 2 y ^ { - 3 / 4 }$$

A) $\frac { 3 y ^ { 2 } - 2 } { y ^ { 3 / 4 } }$
B) $\frac { 3 y ^ { 1 / 2 } - 1 } { 2 y ^ { 3 / 4 } }$
C) $\frac { 3 y ^ { 2 } - 1 } { 2 y ^ { 3 / 4 } }$
D) $\frac { 3 y ^ { 1 / 2 } - 2 } { y ^ { 3 / 4 } }$

Question 27

Evaluate if possible.
$\sqrt { \frac { 121 } { 225 } }$

A) $\frac { 11 } { 16 }$
B) $\frac { 15 } { 28 }$
C) $\frac { 4 } { 5 }$
D) $\frac { 11 } { 15 }$

Question 28

Evaluate the numerical expression.
$8 ^ { 4 / 3 }$

A) 32
B) 16
C) 128
D) 64

Question 29

Evaluate if possible.
$- \sqrt { 225 }$

A) not a real number
B) 15
C) $- 15$
D) $- 112$

Question 30

Factor out the requested common factor.
The radius needed to create a sphere with a given volume $\mathrm { V }$ can be approximated by the equation $\mathrm { r } = 0.62$ $( V ) ^ { 1 / 3 }$ . Find the radius of a sphere with a volume of 125 cubic meters. Round the answer to the nearest hundredth.

A) $25.83$ meters
B) $6.93$ meters
C) $4.26$ meters
D) $3.1$ meters

Question 31

Evaluate if possible.
$\sqrt { 0.0009 }$

A) $0.03$
B) $0.0003$
C) $0.3$
D) $0.003$

Question 32

Factor out the requested common factor.
The radius required for a cone to have a volume $\mathrm { V }$ and a height $\mathrm { h }$ is given by the equation
$$\mathrm { r } = \left( \frac { 3 \mathrm {~V} } { \pi \mathrm { h } } \right) ^ { 1 / 2 }$$
Find the necessary radius to have a cone with the $\mathrm { V } = 1256$ cubic inches and $h = 3$ inches. Use $\pi \approx 3.14$ and round the nearest whole number.

A) 25 inches
B) 20 inches
C) 195 inches
D) 200 inches

Question 33

Evaluate the numerical expression.
$16 ^ { 5 / 4 }$

A) 128
B) 32
C) 256
D) 512

Question 34

Evaluate if possible.
$\sqrt { 100 } - \sqrt { 81 }$

A) 19
B) $\sqrt { 19 }$
C) $- 1$
D) 1

Question 35

Evaluate the numerical expression.
$$9 ^ { 3 / 2 } + 8 ^ { 4 / 3 }$$

A) $24 \frac { 1 } { 6 }$
B) 82
C) 43
D) $48 \frac { 1 } { 6 }$

Question 36

Write the expression as one fraction with positive exponents.
$$3 y ^ { 3 / 4 } + 3 y ^ { - 1 / 4 }$$

A) $\frac { 3 y ^ { 1 / 2 } + 3 } { y ^ { 1 / 4 } }$
B) $\frac { 3 y ^ { 1 / 2 } + 1 } { 3 y ^ { 1 / 4 } }$
C) $\frac { 3 y + 3 } { y ^ { 1 / 4 } }$
D) $\frac { 3 y + 1 } { 3 y ^ { 1 / 4 } }$

Question 37

Evaluate if possible.
$$\sqrt { 25 } + \sqrt { 169 }$$

A) 16
B) $\sqrt { 194 }$
C) 18
D) 194

Question 38

Evaluate the numerical expression.
$243 ^ { 4 / 5 }$

A) 81
B) 6561
C) 19,683
D) 2187

Question 39

Factor out the requested common factor.
Factor out $2 x$ from $10 x ^ { 7 / 5 } - 6 x ^ { 8 / 5 }$

A) $2 x \left( 5 x ^ { 2 / 5 } - 3 x ^ { 3 / 5 } \right)$
B) $2 \times \left( 5 x ^ { 6 / 5 } - 3 x ^ { 7 / 5 } \right)$
C) $2 \times \left( 5 ^ { 7 / 5 } - 3 ^ { 8 / 5 } \right)$
D) $2 \times \left( 5 ^ { 6 / 5 } - 3 ^ { 7 / 5 } \right)$

Question 40

Write the expression as one fraction with positive exponents.
$$3 y ^ { 4 / 3 } + 2 y ^ { - 1 / 3 }$$

A) $\frac { 3 y + 1 } { 2 y 1 / 3 }$
B) $\frac { 3 y ^ { 5 / 3 } + 1 } { 2 y ^ { 1 / 3 } }$
C) $\frac { 3 y ^ { 5 / 3 } + 2 } { y ^ { 1 / 3 } }$
D) $\frac { 3 y + 2 } { y ^ { 1 / 3 } }$

Question 41

Find the domain of the given function.
$$f ( x ) = \sqrt { 21 x - 7 }$$

A) all real numbers $x$ where $x >$
B) all real numbers $x$ where $x > - \frac { 1 } { 7 }$
C) all real numbers $x$ where $x \geq - 7$
D) all real numbers $x$ where $x \geq -$

Question 42

Assume that variables represent positive real numbers. Replace the radical with rational exponents
$\sqrt [ 7 ] { 3 y }$

A) $( 3 y ) ^ { 7 / 2 }$
B) $3 y ^ { 1 / 7 }$
C) $( 3 y ) ^ { 1 / 7 }$
D) $\frac { 3 } { 7 } y ^ { 1 / 7 }$

Question 43

Assume that variables represent positive real numbers. Replace the radical with rational exponents
$\sqrt [ 5 ] { \mathrm { y } }$

A) $\frac { y ^ { 1 / 2 } } { 5 }$
B) $y ^ { 1 / 5 }$
C) $\frac { y } { 5 }$
D) $y ^ { 5 / 2 }$

Question 44

Find the requested function value for the given function. Round the answer to the nearest tenth when necessary.
If $f ( x ) = \sqrt { 2 x - 8 }$ , find the value of $f ( 7 )$ .

A) 14
B) $3.7$
C) $2.4$
D) 6

Question 45

Assume that variables represent positive real numbers. Replace the radical with rational exponents
$\sqrt { \sqrt [ 9 ] { y } }$

A) $\mathrm { y } ^ { 9 / 2 }$
B) $\mathrm { y } ^ { 9 }$
C) $y ^ { 2 / 9 }$
D) $\mathrm { y } ^ { 1 / 18 }$

Question 46

Find the domain of the given function.
$f(x)=\sqrt{x-1}$
A)
B)
C)
D)

Question 47

Assume that variables represent positive real numbers. Replace the radical with rational exponents
$$\sqrt [ 5 ] { ( x + y ) ^ { 3 } }$$

A) $\left( x ^ { 3 } + y ^ { 3 } \right) ^ { 1 / 5 }$
B) $( x + y ) ^ { 3 / 5 }$
C) $\left( x ^ { 5 } + y ^ { 5 } \right) ^ { 1 / 3 }$
D) $( x + y ) ^ { 5 / 3 }$

Question 48

Find the requested function value for the given function. Round the answer to the nearest tenth when necessary.
If $f ( x ) = \sqrt { 2 x - 2 }$ , find the value of $f ( 3 )$ .

A) 6
B) 4
C) 2
D) $2.4$

Question 49

Find the requested function value for the given function. Round the answer to the nearest tenth when necessary.
If $f ( x ) = \sqrt { 2 x - 9 }$ , find the value of $f ( 29 )$ .

A) 7
B) $7.6$
C) 58
D) 49

Question 50

Assume that variables represent positive real numbers. Replace the radical with rational exponents
$( \sqrt [ 4 ] { 2 x } ) ^ { 3 }$

A) $( 6 \times ) ^ { 1 / 4 }$
B) $\left( 2 x ^ { 3 } \right) ^ { 1 / 4 }$
C) $( 2 \mathrm { x } ) ^ { 1 / 12 }$
D) $( 2 x ) ^ { 3 / 4 }$

Question 51

Find the domain of the given function.
$$f ( x ) = \sqrt { 5 x + 25 }$$

A) all real numbers $x$ where $x \geq 5$
B) all real numbers $x$ where $x > 5$
C) all real numbers $x$ where $x \geq - 5$
D) all real numbers $x$ where $x > \frac { 1 } { 5 }$

Question 52

Simplify. Assume that variables represent positive real numbers.
$\sqrt [ 6 ] { ( - 3 ) ^ { 6 } }$

A) 3
B) 18
C) $- 3$
D) $- 18$

Question 53

Simplify. Assume that variables represent positive real numbers.
$3 x ^ { 5 / 7 }$

A) $\sqrt [ 5 ] { 3 x ^ { 7 } }$
В) $\sqrt [ 7 ] { 243 x ^ { 5 } }$
C) $\sqrt [ 7 ] { 3 x ^ { 5 } }$
D) $3 \sqrt [ 7 ] { \mathrm { x } ^ { 5 } }$

Question 54

Simplify. Assume that variables represent positive real numbers.
$\sqrt [ 3 ] { ( - 7 ) ^ { 3 } }$

A) 21
B) 7
C) $- 21$
D) $- 7$

Question 55

Find the domain of the given function.
$$f ( x ) = \sqrt { 3.5 x + 3 }$$

A) all real numbers $x$ where $x > -$
B) all real numbers $x$ where $x > - \frac { 3 } { 35 }$
C) all real numbers $x$ where $x \geq -$
D) all real numbers $x$ where $x \geq - \frac { 6 } { 7 }$

Question 56

Simplify. Assume that variables represent positive real numbers.
$$\sqrt [ 4 ] { - 81 x ^ { 8 } y ^ { 12 } }$$

A) $- 81 x ^ { 2 } y ^ { 8 }$
B) not a real number
C) $- 3 x ^ { 2 } y ^ { 3 }$
D) $- 3 x ^ { 4 } y ^ { 8 }$

Question 57

Simplify. Assume that variables represent positive real numbers.
$\sqrt [ 3 ] { \mathrm { x } ^ { 3 } \mathrm { y } ^ { 6 } }$

A) $y ^ { 3 }$
B) $x y ^ { 2 }$
C) $x y ^ { 3 }$
D) $x ^ { 3 } y ^ { 2 }$

Question 58

Find the domain of the given function.
$f(x)=\sqrt{3 x+6}$
A)
B)
C)
D)

Question 59

Simplify. Assume that variables represent positive real numbers.
$( 3 x ) ^ { 4 / 9 }$

A) $\sqrt [ 4 ] { 3 x ^ { 9 } }$
B) $\sqrt [ 9 ] { 3 x ^ { 4 } }$
C) $\sqrt [ 9 ] { 81 x ^ { 4 } }$
D) $3 \sqrt [ 9 ] { x ^ { 4 } }$

Question 60

Find the requested function value for the given function. Round the answer to the nearest tenth when necessary.
If $f ( x ) = \sqrt { 2.5 x + 4 }$ , find the value of $f ( 10 )$

A) 29
B) 31
C) $5.7$
D) $5.4$

Question 61

Simplify. Assume that variables represent positive real numbers.
$\sqrt { 9 x ^ { 6 } y ^ { 18 } z ^ { 12 } }$

A) $9 x ^ { 3 } y ^ { 16 } z ^ { 10 }$
B) $3 x ^ { 4 } y ^ { 16 } z ^ { 10 }$
C) $9 x ^ { 4 } y ^ { 9 } z ^ { 10 }$
D) $3 x ^ { 3 } y ^ { 9 } z ^ { 6 }$

Question 62

Simplify. Assume that variables represent positive real numbers.
$\sqrt [ 5 ] { 1024 x ^ { 10 } y ^ { 20 } }$

A) $4 x ^ { 5 } y ^ { 15 }$
B) $4 x ^ { 2 } y ^ { 4 }$
C) $1024 x ^ { 2 } y ^ { 4 }$
D) $1024 x ^ { 5 } y ^ { 15 }$