Deck 7: Radicals, Radical Functions, and Rational Exponents

Question 1

Find the square root if it is a real number, or state that the expression is not a real number.
$\sqrt { 100 - 200 }$

A) $- 100$
B) 10
C) not a real number
D) $- 10$

Answer 1

C

Question 2

Find the square root if it is a real number, or state that the expression is not a real number.
$\sqrt { - 64 }$

A) 4096
B) $\frac { 8 } { 64 }$
C) 8
D) not a real number

Answer 2

D

Question 3

Find the square root if it is a real number, or state that the expression is not a real number.
$\sqrt { 4 }$

A) $\frac { 1 } { 4 }$
B) not a real number
C) 16
D) 2

Answer 3

D

Question 4

Find the domain of the square root function. Then use the domain to choose the correct graph of the function.
$$f ( x ) = \sqrt { 4 - x }$$

A) domain of $f : [ 4 , \infty )$

B) domain of $\mathrm { f } : ( \infty , 4 ]$

C) domain of $f : ( * , - 4 ]$

D) domain of $f : [ - 4 , \infty )$

Question 5

Find the square root if it is a real number, or state that the expression is not a real number.
$- \sqrt { 36 }$

A) $- 18$
B) not a real number
C) 6
D) $- 6$

Question 6

Find the function value indicated for the function. If necessary, round to two decimal places. If the function value is not a real number and does not exist, state so.
Evaluate $f ( x ) = \sqrt { x - 25 }$ for $f ( 1 )$

A) $- \sqrt { 24 }$
B) not a real number
C) 24
D) $- 4$

Question 7

Find the function value indicated for the function. If necessary, round to two decimal places. If the function value is not a real number and does not exist, state so.
$$\text { Evaluate } f ( x ) = - \sqrt { 2 x + 5 } \text { for } f ( 2 )$$

A) -1.73
B) 3
C) not a real number
D) -3

Question 8

Find the square root if it is a real number, or state that the expression is not a real number.
$$\sqrt { 144 + 25 }$$

A) $\sqrt { 119 }$
B) 13
C) 169
D) 17

Question 9

Find the square root if it is a real number, or state that the expression is not a real number.
$\sqrt { \frac { 49 } { 256 } }$

A) $\frac { 1 } { 2 }$
B) $\frac { 7 } { 17 }$
C) $\frac { 3 } { 16 }$
D) $\frac { 7 } { 16 }$

Question 10

Find the function value indicated for the function. If necessary, round to two decimal places. If the function value is not a real number and does not exist, state so.
Evaluate $g ( x ) = \sqrt { x - 5 }$ for $g ( 30 )$

A) 5
B) not a real number
C) $3.24$
D) $5.48 \mathrm {~g}$

Question 11

Find the domain of the square root function. Then use the domain to choose the correct graph of the function.
$$f ( x ) = \sqrt { x - 2 }$$

A) domain of $\mathrm { f: } [ 2 , \infty )$

B) domain of $f : [ - 2 , \infty )$

C) domain of $f : ( - , - 2 ]$

D) domain of $f : ( * , 2 ]$

Question 12

Find the function value indicated for the function. If necessary, round to two decimal places. If the function value is not a real number and does not exist, state so.
$$\text { Evaluate } t ( x ) = - \sqrt { x + 6 } \text { for } t ( - 2 )$$

A) -1.41
B) 2
C) -2
D) not a real number

Question 13

Find the function value indicated for the function. If necessary, round to two decimal places. If the function value is not a real number and does not exist, state so.
$$\text { Evaluate } f ( x ) = \sqrt { - x + 34 } \text { for } f ( - 2 )$$

A) 6
B) 2.45
C) not a real number
D) 5.66

Question 14

Find the square root if it is a real number, or state that the expression is not a real number.
$\sqrt { 30 - 5 }$

A) 25
B) $\sqrt { 30 } + \sqrt { 5 }$
C) 5
D) not a real number

Question 15

Find the function value indicated for the function. If necessary, round to two decimal places. If the function value is not a real number and does not exist, state so.
$z ( x ) = \sqrt { ( x + 7 ) ^ { 2 } }$ for $z ( - 3 )$

A) 16
B) 4
C) not a real number
D) 58

Question 16

Find the function value indicated for the function. If necessary, round to two decimal places. If the function value is not a real number and does not exist, state so.
Evaluate $m ( x ) = - \sqrt { x - 2 }$ for $m ( 27 )$

A) $m ( - 5 )$
B) 5
C) not a real number
D) $- 5$

Question 17

Find the function value indicated for the function. If necessary, round to two decimal places. If the function value is not a real number and does not exist, state so.
$$\text { Evaluate } c ( x ) = \sqrt { x + 6 } \text { for } c ( - 2 )$$

A) 2
B) not a real number
C) -2
D) 1.41

Question 18

Find the square root if it is a real number, or state that the expression is not a real number.
- 0.49

A) not a real number
B) 0.07
C) -0.7
D) -0.07

Question 19

Find the domain of the square root function. Then use the domain to choose the correct graph of the function.
$$f ( x ) = \sqrt { x + 3 }$$

A) domain of $f : [ - 3 , \infty )$

B) domain of $f : [ 3 , \infty )$

C) domain of $\mathrm { f } : ( \infty , - 3 ]$

D) domain of $\mathrm { f } : ( \infty , 3 ]$

Question 20

Find the square root if it is a real number, or state that the expression is not a real number.
$\sqrt { \frac { 1 } { 81 } }$

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

Question 21

Find the domain of the square root function. Then use the domain to choose the correct graph of the function.
$$f ( x ) = \sqrt { 9 x - 36 }$$

A) domain of $f : ( * , 4 ]$

B) domain of $\mathrm { f } : [ - 4 , \infty )$

C) domain of $f : [ 4 , \infty )$

D) domain of $f : ( x , - 4 ]$

Question 22

Find the indicated function value for the function.
Evaluate $m ( x ) = - \sqrt [ 3 ] { 2 x - 2 }$ for $m ( 5 )$

A) $\sqrt [ 3 ] { 2 }$
B) 2
C) 3
D) $- 2$

Question 23

Find the cube root.
$$- \sqrt [ 3 ] { 729 }$$

A) 9
B) 81
C) $- 729$
D) $- 9$

Question 24

Solve the problem.
The formula $\mathrm { v } = \sqrt { 2.5 \mathrm { r } }$ models the safe maximum speed, $\mathrm { v }$ , in miles per hour, at which a car can travel on a curved road with radius of curvature, $r$ , in feet. A highway crew measures the radius of curvature at an exit ramp as 360 feet. What is the maximum safe speed?

A) 30 mph
B) 36 mph
C) 27 mph
D) 35 mph

Question 25

Find the cube root.
$$\sqrt [ 3 ] { - 8 }$$

A) -2
B) 4
C) ±2
D) not a real number

Question 26

Simplify the expression.
$$\sqrt { ( x - 5 ) ^ { 2 } }$$

A) $| x - 5 |$
B) $- 5 \sqrt { x }$
C) $x - 5$
D) $x ^ { 2 } - 10 x + 25$

Question 27

Simplify the expression.
$$\sqrt { x ^ { 2 } - 14 x + 49 }$$
B) $x - 7$
C) $| ( x - 7 ) ( x + 7 ) |$
D) $| x - 7 |$

Question 28

Simplify the expression.
$\sqrt { ( - 5 ) ^ { 2 } }$
B) $- \sqrt { 5 }$
C) 5
D) 25

A) $- 5$
B) $- \sqrt { 5 }$
C) 5

Question 29

Find the indicated function value for the function.
$$\text { Evaluate } \mathrm { f } ( \mathrm { x } ) = \sqrt [ 3 ] { \mathrm { x } + 7 } \text { for } \mathrm { f } ( 118 )$$

A) 5
B) 5, -5
C) 12
D) 125

Question 30

Simplify the expression.
$- \sqrt { 49 x ^ { 8 } }$

A) $\left| 7 x ^ { 4 } \right|$
B) $- \left| 49 x ^ { 4 } \right|$
C) $- 7 x ^ { 8 }$
D) $- \left| 7 x ^ { 4 } \right|$

Question 31

Find the cube root.
$$\sqrt [ 3 ] { 125 }$$

A) 11
B) 5
C) 25
D) ±5

Question 32

Find the indicated function value for the function.
$$\text { Evaluate } f ( x ) = \sqrt [ 3 ] { x - 27 } \text { for } f ( 0 )$$

A) 3
B) not a real number
C) -3
D) 0

Question 33

Simplify the expression.
$$- \sqrt { x ^ { 2 } - 16 x + 64 }$$

A) $- | x - 8 |$
B) $- x + 8$
C) $| x - 8 |$
D) $- x - 8$

Question 34

Simplify the expression.
$\sqrt { 5 ^ { 2 } }$

A) $\sqrt { 10 }$
B) $\frac { 5 } { 2 }$
C) 25
D) 5

Question 35

Find the cube root.
$\sqrt [ 3 ] { \frac { 1 } { 64 } }$

A) $\frac { 1 } { 4 }$
B) 4
C) $\frac { 1 } { \sqrt [ 3 ] { 4 } }$
D) $\frac { 1 } { 16 }$

Question 36

Find the indicated function value for the function.
$$\text { Evaluate } f ( x ) = \sqrt [ 3 ] { x + 3 } \text { for } f ( - 11 )$$

A) 2
B) -8
C) -2
D) not a real number

Question 37

Solve the problem.
The formula $\mathrm { v } = \sqrt { 20 \mathrm {~L} }$ can be used to estimate the speed of a car, $\mathrm { v }$ , in miles per hour, based on the length, L, in feet, of its skid marks upon sudden braking on a dry asphalt road. If a car is involved in an accident and its skid marks measure 45 feet, at what estimated speed was the car traveling when it applied its brakes just prior to the accident?

A) 25 mph
B) 40 mph
C) 30 mph
D) 35 mph

Question 38

Solve the problem.
The average height of a boy in the United States, from birth through 60 months, can be modeled by y = 2.9x + 20.1 where y is the average height, in inches, of boys who are x months of age. What would be the expected difference in height between a child 25 months of age and a child 16 months of age?

A) 43.1 in.
B) 14.5 in.
C) 4.9 in.
D) 2.9 in.

Question 39

Find the domain of the square root function. Then use the domain to choose the correct graph of the function.
$$f ( x ) = \sqrt { 64 - 8 x }$$

A) domain of $f : [ 8 , \infty )$

B) domain of $\mathrm { f } : [ - 8 , \infty )$

C) domain of $f : ( \infty , - 8 ]$

D) domain of $\mathrm { f } : ( \infty , 8 ]$

Question 40

Simplify the expression.
$\sqrt { 4 x ^ { 8 } }$

A) $2 x ^ { 8 }$
B) $4 x ^ { 4 }$
C) $| 2 x |$
D) $2 x ^ { 4 }$

Question 41

Simplify the expression. Include absolute value bars where necessary.
$\sqrt [ 3 ] { - 64 x ^ { 3 } }$

A) $4 x$
B) $- | - 4 | x$
C) $| - 4 | x$
D) $- 4 x$

Question 42

Find the indicated root, or state that the expression is not a real number.
$\sqrt [ 4 ] { 81 }$

A) 9
B) 12
C) 3
D) $- 3$

Question 43

Find the indicated root, or state that the expression is not a real number.
$\sqrt [ 3 ] { - 27 }$

A) $- 3$
B) 3
C) not a real number
D) $- \sqrt [ 3 ] { 3 }$

Question 44

Simplify the expression. Include absolute value bars where necessary.
$\sqrt [ 3 ] { - 8 ( x - 3 ) ^ { 3 } }$
B) $- 2 x - 6$
C) $- 8 x - 3$
D) $| - 2 x + 6 |$

Question 45

Simplify the expression. Include absolute value bars where necessary.
$\sqrt [ 4 ] { ( - 4 ) ^ { 4 } }$
B) $| - 256 |$
C) $- | - 4 |$
D) 4

A) $- 4$

Question 46

Find the indicated root, or state that the expression is not a real number.
$- \sqrt [ 4 ] { 16 }$

A) not a real number
B) $- 2$
C) $- 4$
D) 2

Question 47

Find the indicated function value for the function.
Evaluate $g ( x ) = - \sqrt [ 3 ] { - 2 x - 70 }$ for $g ( - 3 )$

A) not a real number
B) $- 4$
C) $\sqrt [ 3 ] { 4 }$
D) 4

Question 48

Find the indicated root, or state that the expression is not a real number.
$\sqrt [ 6 ] { - 1 }$

A) 1
B) $- 1$
C) not a real number
D) 0

Question 49

Simplify the expression. Include absolute value bars where necessary.
$\sqrt [ 5 ] { ( - 2 ) ^ { 5 } }$

A) $- 2$
B) $| - 2 |$
C) 2
D) $( - 2 ) ^ { 5 }$

Question 50

Find the indicated root, or state that the expression is not a real number.
$\sqrt [ 3 ] { 64 }$

A) $\sqrt [ 3 ] { 4 }$
B) 12
C) 4
D) $- 4$

Question 51

Simplify the expression. Include absolute value bars where necessary.
$$\sqrt [ 6 ] { ( x + 2 ) ^ { 6 } }$$

A) $| x + 2 |$
B) $| x | + 2$
C) $\pm x + 2$
D) $x + 2$

Question 52

Find the indicated root, or state that the expression is not a real number.
$\sqrt [ 5 ] { - 1 }$

A) $- \frac { 1 } { 5 }$
B) not a real number
C) 1
D) $- 1$

Question 53

Find the indicated function value for the function.
Evaluate $f ( x ) = - \sqrt [ 3 ] { 64 x - 1 }$ for $f ( 0 )$

A) not a real number
B) 1
C) $- 1$
D) $- 3$

Question 54

Find the indicated root, or state that the expression is not a real number.
$$- \sqrt [ 4 ] { - 256 }$$

A) not a real number
B) $- 4$
C) $- 16$
D) 4

Question 55

Simplify the expression. Include absolute value bars where necessary.
$\sqrt [ 8 ] { x ^ { 8 } }$

A) $- x$
B) $x$
C) $| x |$
D) $- | x |$

Question 56

Simplify the expression. Include absolute value bars where necessary.
$$\sqrt [ 4 ] { ( x + 81 ) ^ { 4 } }$$

A) $| x + 3 |$
B) $x + 3$
C) $x + 81$
D) $| x + 81 |$

Question 57

Simplify the expression. Include absolute value bars where necessary.
$\sqrt [ 7 ] { g ^ { 7 } }$

A) $\mathrm { g }$
B) $- g$
C) $\sqrt [ 7 ] { \mathrm { g } }$
D) $g ^ { 7 }$

Question 58

Find the indicated root, or state that the expression is not a real number.
$$\sqrt [ 4 ] { - 256 }$$

A) not a real number
B) -16
C) -4
D) 4

Question 59

Find the indicated root, or state that the expression is not a real number.
$$- \sqrt [ 3 ] { - 125 }$$

A) not a real number
B) 5
C) $- 5$
D) 125

Question 60

Find the indicated root, or state that the expression is not a real number.
$- \sqrt [ 3 ] { 8 }$

A) $- 2$
B) 2
C) $- \sqrt [ 3 ] { 2 }$
D) not a real number

Question 61

Rewrite the expression with a rational exponent.
$\sqrt { 6 }$

A) $\frac { 1 } { 6 ^ { 2 } }$
B) $6 ^ { 1 / 2 }$
C) $6 ^ { 2 }$
D) $\frac { 1 } { 2 } \cdot 6$

Question 62

Use radical notation to rewrite the expression. Simplify, if possible.
$\left( 2 x y ^ { 4 } \right) ^ { 1 / 5 }$

A) $\sqrt [ 5 ] { 2 x y ^ { 4 } }$
B) $\frac { 1 } { 2 x y ^ { 4 } }$
C) $y \sqrt [ 5 ] { 2 x }$
D) $\frac { 1 } { \sqrt [ 5 ] { 2 x y ^ { 4 } } }$

Question 63

Use radical notation to rewrite the expression. Simplify, if possible.
$16 ^ { 1 / 4 }$

A) 4
B) $\frac { 1 } { 2 }$
C) 2
D) $- \sqrt [ 4 ] { 2 }$

Question 64

Rewrite the expression with a rational exponent.
$\sqrt [ 5 ] { 9 x ^ { 6 } y }$

A) $\frac { 1 } { 5 } \cdot \left( 9 x ^ { 6 } y \right)$
B) $\left( 9 x ^ { 6 } y \right) ^ { 1 / 5 }$
C) $\left( 9 x ^ { 6 } y \right) ^ { 5 }$
D) $\frac { 1 } { \left( 9 x ^ { 6 } y \right) ^ { 5 } }$

Question 65

Use radical notation to rewrite the expression. Simplify, if possible.
$( x y ) ^ { 1 / 5 }$

A) $\sqrt [ 5 ] { x y }$
B) $\frac { \sqrt [ 5 ] { x y } } { 5 }$
C) $\frac { 1 } { \sqrt [ 5 ] { x y } }$
D) $\sqrt { \mathrm { x } ^ { 5 } \mathrm { y } ^ { 5 } }$

Question 66

Use radical notation to rewrite the expression. Simplify, if possible.
$( x y ) ^ { 8 / 9 }$

A) $\sqrt [ 9 ] { \mathrm { xy } ^ { 8 } }$
B) $\sqrt [ 9 ] { ( x y ) ^ { 8 } }$
C) $\frac { ( x y ) ^ { 8 } } { 9 }$
D) $\sqrt [ 8 ] { ( x y ) ^ { 9 } }$

Question 67

Rewrite the expression with a rational exponent.
$\sqrt [ 3 ] { 45 }$

A) $\frac { 1 } { 3 } \cdot 45$
B) $45 ^ { 3 }$
C) $\frac { 1 } { 45 ^ { 3 } }$
D) $45 ^ { 1 / 3 }$

Question 68

Use radical notation to rewrite the expression. Simplify, if possible.
$( - 64 ) ^ { 4 / 3 }$

A) not a real number
В) 256
C) 4096
D) $- 256$

Question 69

Use radical notation to rewrite the expression. Simplify, if possible.
$125 ^ { 4 / 3 }$

A) 78,125
B) 3125
C) 15,625
D) 625

Question 70

Use radical notation to rewrite the expression. Simplify, if possible.
$144 ^ { 1 / 2 }$

A) 72
B) 12
C) 6
D) $\frac { 1 } { 6 }$

Question 71

Use radical notation to rewrite the expression. Simplify, if possible.
$( 125 ) ^ { 2 / 3 }$

A) 5
В) $\frac { 250 } { 3 }$
C) 10
D) 25

Question 72

Use radical notation to rewrite the expression. Simplify, if possible.
$32 ^ { 4 / 5 }$

A) 512
B) 16
C) 128
D) 256

Question 73

Use radical notation to rewrite the expression. Simplify, if possible.
$625 ^ { 5 / 4 }$

A) 390,625
В) 78,125
C) 3125
D) $1,953,125$

Question 74

Use radical notation to rewrite the expression. Simplify, if possible.
$- 16 ^ { 1 / 4 }$

A) $- \sqrt [ 4 ] { 2 }$
B) $- \frac { 1 } { 2 }$
C) $- 2$
D) 2

Question 75

Rewrite the expression with a rational exponent.
$\sqrt [ 4 ] { 8 }$

A) $8 ^ { 4 }$
B) $\frac { 1 } { 8 ^ { 4 } }$
C) $\frac { 1 } { 4 } \cdot 8$
D) $8 ^ { 1 / 4 }$

Question 76

Use radical notation to rewrite the expression. Simplify, if possible.
$$( - 125 ) ^ { 2 / 3 }$$

A) 25
B) $- \frac { 1 } { 5 }$
C) $- 5$
D) $- 25$

Question 77

Use radical notation to rewrite the expression. Simplify, if possible.
$( - 27 ) ^ { 1 / 3 }$

A) $- 3 \sqrt [ 3 ] { 9 }$
B) 3
C) $- 3$
D) $- \frac { 1 } { 3 }$

Question 78

Rewrite the expression with a rational exponent.
$\sqrt [ 7 ] { x y ^ { 2 } }$

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

Question 79

Use radical notation to rewrite the expression. Simplify, if possible.
$$16 ^ { 3 / 4 } + 64 ^ { 2 / 3 }$$

A) $113 \frac { 1 } { 3 }$
B) 24
C) 25
D) $54 \frac { 2 } { 3 }$

Question 80

Rewrite the expression with a rational exponent.
$\sqrt [ 3 ] { 5 p }$

A) $( 5 p ) ^ { 3 }$
B) $\frac { 1 } { ( 5 p ) ^ { 3 } }$
C) $5 ^ { 1 / 3 } \mathrm { p }$
D) $( 5 p ) ^ { 1 / 3 }$