Deck 7: Radical Expressions and Equations

Question 1

$f ( x ) = \sqrt { x }$ and $g ( x ) = \sqrt [ 7 ] { x }$ are __________ functions.

A) cubic
B) radical
C) quadratic
D) rational
E) linear

Answer 1

radical

Question 2

Simplify the radical expression. Assume a is unrestricted. $\sqrt [ 3 ] { 8 a ^ { 3 } }$

A) $2 a ^ { 3 }$
B) 8a
C) $8 a ^ { 3 }$
D) 2a
E) not real

Answer 2

2a

Question 3

Simplify the expression. Assume that x is unrestricted and use absolute value symbols if necessary. $\sqrt { 36 x ^ { 2 } }$

A) 5x
B) $6 | x |$
C) 6x
D) 4x
E) $36 | x |$

Answer 3

$6 | x |$

Question 4

Find the value given that $g ( x ) = \sqrt [ 3 ] { x - 4 }$ . $g ( 5 ) = ?$

A) 4
B) 2
C) 5
D) 7
E) 1

Question 5

Evaluate the square root, if possible, without using a calculator. $\sqrt { 100 }$

A) 8
B) 10
C) 9
D) 11
E) not real

Question 6

Evaluate the square root, if possible, without using a calculator. $\sqrt { - 9 }$

A) 2
B) 3
C) -5
D) $5 x - 4$
E) not real

Question 7

Simplify the cube root. $\sqrt [ 3 ] { - 512 }$

A) 11
B) -7
C) 8
D) -8
E) 7

Question 8

Simplify the cube root. $\sqrt [ 3 ] { - 64 x ^ { 6 } }$

A) $- 4 x ^ { 2 }$
B) $5 x ^ { 3 }$
C) $4 x ^ { 2 }$
D) - $5 x ^ { 3 }$
E) -4x

Question 9

Find the domain and range of the function $f ( x )$ . $f ( x ) = \sqrt [ 3 ] { x } + 2$

A) $D = ( - \infty , 2 ) ; R = ( - \infty , 2 )$
B) $D = [ 2 , \infty ) ; R = [ - \infty , 2 )$
C) $D = ( 2 , \infty ) ; R = ( 2 , \infty )$
D) $D = [ 1 , \infty ) ; R = [ - \infty , 1 )$
E) $D = ( - \infty , \infty ) ; R = ( - \infty , \infty )$

Question 10

Evaluate the radical expression, if possible, without using a calculator. $\sqrt [ 4 ] { - 625 }$

A) 5
B) -5
C) -625
D) 625
E) not real

Question 11

The graph of a square root function $f$ is shown below.
Find the value(s) of x for which $f ( x ) = 1$ .

A) x=3
B) x=6
C) x=5
D) $x = 4$
E) $x = 2$

Question 12

Find the value given that $f ( x ) = \sqrt { x - 1 }$ . $f ( 17 ) = ?$

A) 4
B) -4
C) 1
D) $- 1$
E) -2

Question 13

In the expression $\sqrt [ 3 ] { 27 x ^ { 6 } }$ , the index is 3 and $27 x ^ { 6 }$ is the __________.

A) power
B) base
C) radicand
D) index
E) argument

Question 14

Simplify $\sqrt { ( - 10 b ) ^ { 2 } }$ . Assume all variables are unrestricted.

A) $- 10 | b |$
B) $10 | b |$
C) -10b
D) 10b

Question 15

Find the domain and range of the function $f ( x )$ . $f ( x ) = \sqrt { x } + 1$

A) $D = [ 0 , \infty ) ; R = [ - 2 , \infty )$
B) $D = [ - \infty , - 2 ) ; R = ( - \infty , \infty )$
C) $D = [ 0 , \infty ) ; R = ( 1 , \infty )$
D) $D = [ - \infty , - 2 ) ; R = ( - 2 , \infty )$
E) $D = [ 0 , \infty ) ; R = [ 1 , \infty )$

Question 16

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

A) $[ 0 , \infty )$
B) $[ - 7 , \infty )$
C) $[ 7 , \infty )$
D) $( - \infty , 7 ) \cup ( 7 , \infty )$

Question 17

Simplify: $\sqrt [ 3 ] { - \frac { 27 x ^ { 9 } } { 64 y ^ { 6 } } }$

A) $- \frac { 3 x ^ { 3 } } { 4 y ^ { 2 } }$
B) not a real number
C) $- \frac { 3 x ^ { 3 } } { 16 y ^ { 4 } }$
D) $- \frac { 9 x ^ { 3 } } { 16 y ^ { 2 } }$

Question 18

Simplify the cube root. $\sqrt [ 3 ] { - 27 a ^ { 3 } }$

A) -3a
B) $- 3 a ^ { 3 }$
C) -6a
D) 3|a|
E) 6|a|

Question 19

Evaluate the square root, if possible, without using a calculator. $- \sqrt { 169 }$

A) 12
B) -12
C) 15
D) -13
E) not real

Question 20

Evaluate the square root, if possible, without using a calculator. $\sqrt { \frac { 1 } { 81 } }$

A) $\frac { 1 } { 9 }$
B) $\frac { 1 } { 13 }$
C) $\frac { 1 } { 11 }$
D) $\frac { 1 } { 10 }$
E) not real

Question 21

The number 121 has two square roots. The positive or __________ square root of 121 is 11.

Question 22

Simplify the cube root.

Question 23

Simplify: $- \sqrt [ 3 ] { - 125 x ^ { 9 } }$

A) $- 4 x ^ { 3 }$
B) $- 5 x ^ { 3 }$
C) $5 x ^ { 3 }$
D) not a real number

Question 24

Simplify $- \sqrt [ 4 ] { 16 m ^ { 24 } }$ . Assume all variables represent positive real numbers.

A) $- 2 m ^ { 6 }$
B) $- 4 m ^ { 6 }$
C) $- 4 m ^ { 12 }$
D) not a real number

Question 25

Simplify $\sqrt { x ^ { 2 } - 14 x + 49 }$ . Assume all variables are unrestricted and use absolute value symbols when necessary.

A) x-7
B) $( x - 7 ) ^ { 2 }$
C) 7|x|
D) | x-7|

Question 26

The __________ root of 125 is 5 because

Question 27

Evaluate the square root, if possible, without using a calculator.

Question 28

Evaluate the square root, if possible, without using a calculator.

Question 29

Find the domain of $f ( x ) = \sqrt { 8 - x }$ .

A) $[ 8 , \infty )$
B) $( - \infty , 8 ]$
C) $[ - 8 , \infty )$
D) $( - \infty , - 8 ]$

Question 30

In the expression , the __________ is and is the __________.

Question 31

Evaluate the square root, if possible, without using a calculator.

Question 32

Find f (6) for $f ( x ) = \sqrt { x - 8 }$ .

A) 2
B) -2
C) 3
D) not a real number

Question 33

Find the value given that . __________

Question 34

and are __________ functions.

Question 35

Find the value given that . __________

Question 36

Find the domain of $f ( x ) = \sqrt { x - 1 }$ .

A) $[ 1 , \infty )$
B) $( - \infty , 1 ]$
C) $[ 0 , \infty )$
D) $( - \infty , \infty )$

Question 37

Find $f ( - 4 )$ , given $f ( x ) = \sqrt [ 3 ] { x - 23 }$

A) 3
B) -3
C) 2
D) not a real number

Question 38

Simplify the expression. Assume that is unrestricted and use absolute value symbols if necessary.

Question 39

Simplify the cube root.

Question 40

Simplify the radical, if possible.

Question 41

Change the exponential expression to a radical. $\left( x ^ { 5 } + y ^ { 6 } \right) ^ { \frac { 1 } { 2 } }$

A) $\sqrt { x ^ { 10 } + y ^ { 12 } }$
B) $x ^ { 5 } + y ^ { 6 }$
C) $\sqrt { x ^ { 5 } + y ^ { 6 } }$
D) $\sqrt { \left( x ^ { 5 } + y ^ { 6 } \right) ^ { 2 } }$
E) $\sqrt { x ^ { 5 } } + \sqrt { y ^ { 6 } }$

Question 42

Simplify the expression. $\left( - \frac { 8 } { 27 } \right) ^ { - \frac { 4 } { 3 } }$

A) $\frac { 16 } { 81 }$
B) $\frac { 81 } { 16 }$
C) $\frac { 2 } { 3 }$
D) $\left( \frac { 3 } { 2 } \right) ^ { 4 }$
E) $\frac { 3 } { 2 }$

Question 43

Evaluate the expression. $125 ^ { \frac { 2 } { 3 } }$

A) $\left( 125 ^ { \frac { 1 } { 3 } } \right) ^ { 2 }$
B) $5 ^ { 2 }$
C) 5
D) 25
E) $15,625 ^ { \frac { 1 } { 3 } }$

Question 44

Simplify the expression. $25 ^ { - \frac { 3 } { 2 } }$

A) 125
B) - $\frac { 1 } { 125 }$
C) 25
D) $\frac { 1 } { 125 }$
E) -125

Question 45

Evaluate the square root, if possible, without using a calculator.

Question 46

Evaluate the expression. $\frac { 1 } { 3 }$

A) 6
B) 216
C) 18
D) 24
E) not real

Question 47

Simplify the expression. All variables represent positive real numbers. $- \left( 16 x ^ { 2 } \right) ^ { - \frac { 3 } { 2 } }$

A) $- \frac { 1 } { 64 x ^ { 3 } }$
B) $- \frac { 16 } { x ^ { 3 } }$
C) $- \frac { 64 } { x ^ { 3 } }$
D) $- \frac { x ^ { 2 } } { 64 }$
E) $- \frac { x ^ { 3 } } { 64 }$

Question 48

Simplify the expression. $36 ^ { - \frac { 1 } { 2 } }$

A) 6
B) $1 + 9 \log _ { 4 } x - \log _ { 4 } y$
C) 36
D) $- \frac { 1 } { 36 ^ { \frac { 1 } { 2 } } }$
E) -6

Question 49

In the exponential expression $48 ^ { \frac { 5 } { 4 } }$ , 48 is the __________ and $\frac { 5 } { 4 }$ is the __________.

A) radicand, power
B) index, radicand
C) base, exponent
D) index, root
E) power, root

Question 50

Change the radical to an exponential expression. $\sqrt [ 3 ] { 3 k l m }$

A) $( 3 k l m ) ^ { \frac { 1 } { 3 } }$
B) $( 3 \mathrm { klm } ) ^ { 3 }$
C) $3 ( \mathrm { klm } ) ^ { 3 }$
D) $3 ( k l m ) ^ { \frac { 1 } { 3 } }$
E) $3 ^ { \frac { 1 } { 3 } } + k l m ^ { \frac { 1 } { 3 } }$

Question 51

Show that is equal to .

Question 52

Evaluate the expression. $- 27 ^ { \frac { 1 } { 3 } }$

A) -9
B) -6
C) -3
D) $\sqrt [ 3 ] { 27 }$
E) not real

Question 53

Graph . Give the domain and range. Domain:__________ Range:___________

Question 54

Evaluate the expression. $( - 125 ) ^ { \frac { 1 } { 3 } }$

A) -15
B) -10
C) -5
D) -16
E) not real

Question 55

Simplify the cube root.

Question 56

Simplify the expression. All variables represent positive real numbers. $\left( 4 x ^ { 4 } \right) ^ { \frac { 3 } { 2 } }$

A) $\left( 2 x ^ { 2 } \right) ^ { 3 }$
B) $8 \left( x ^ { 4 } \right) ^ { \frac { 3 } { 2 } }$
C) $8 x ^ { 6 }$
D) $8 x ^ { 12 }$
E) $8 x ^ { 2 }$

Question 57

Evaluate the expression. $625 ^ { \frac { 1 } { 4 } }$

A) 2,500
B) 5
C) 1000
D) 25
E) not real

Question 58

Simplify the expression. Write the answer without negative exponents. $7 ^ { - \frac { 1 } { 3 } } 7 ^ { - \frac { 5 } { 3 } }$

A) $\frac { 1 } { 49 }$
B) $7 ^ { \frac { 4 } { 3 } }$
C) $\frac { 1 } { 7 ^ { \frac { 2 } { 3 } } }$
D) $7 ^ { \frac { 2 } { 3 } }$
E) 49

Question 59

Simplify and assume that a 0 and b 0.

Question 60

Simplify the radical expression. Assume is unrestricted.

Question 61

Complete the rule for exponents.

Question 62

Change the exponential expression to a radical.

Question 63

Evaluate the expression.

Question 64

Evaluate the expression.

Question 65

Change the radical to an exponential expression.

Question 66

Evaluate the expression.

Question 67

In the radical expression , is the __________.

Question 68

Evaluate the expression.

Question 69

In the exponential expression , 27 is the __________ and is the __________.

Question 70

Evaluate the expression.

Question 71

Simplify: $\left( 27 x ^ { 3 } y ^ { 3 } \right) ^ { - \frac { 2 } { 3 } }$

A) $\frac { 1 } { 9 x ^ { 2 } y ^ { 2 } }$
B) $\frac { 1 } { 68 x ^ { 2 } y ^ { 2 } }$
C) $\frac { 27 } { x ^ { 2 } y ^ { 2 } }$
D) $- \frac { 9 } { x ^ { 2 } y ^ { 2 } }$

Question 72

Simplify: $16 ^ { \frac { 3 } { 2 } }$

A) $\frac { 1 } { 64 }$
B) -64
C) not a real number
D) 64

Question 73

Evaluate the expression:

Question 74

Simplify the expression. All variables represent positive real numbers.

Question 75

Simplify the expression. Write the answer without negative exponents. $\frac { 4 ^ { \frac { 1 } { 6 } } 4 ^ { \frac { 2 } { 3 } } } { 4 ^ { \frac { 1 } { 2 } } }$

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

Question 76

Perform the indicated operation. $5 ^ { - \frac { 4 } { 3 } } \cdot 5 ^ { \frac { 5 } { 6 } }$

A) $\frac { 1 } { 25 ^ { \frac { 1 } { 2 } } }$
B) $\frac { 1 } { 5 ^ { \frac { 1 } { 2 } } }$
C) $25 ^ { \frac { 1 } { 2 } }$
D) $5 ^ { \frac { 1 } { 2 } }$

Question 77

Simplify:

A) 2
B) 66
C) 16
D) 73

Question 78

Perform the operation. Write the answer without negative exponents. $\frac { 5 ^ { - \frac { 1 } { 5 } } \cdot 5 ^ { \frac { 4 } { 5 } } } { 5 ^ { \frac { 7 } { 5 } } }$

A) $\frac { 1 } { 25 ^ { \frac { 3 } { 5 } } }$
B) $5 ^ { \frac { 4 } { 5 } }$
C) $\frac { 1 } { 5 }$
D) $\frac { 1 } { 5 ^ { \frac { 4 } { 5 } } }$

Question 79

Perform the indicated operation. Assume all variables represent positive numbers. $\left( 9 x ^ { - 12 } \right) ^ { - \frac { 1 } { 2 } }$

A) $- 3 x ^ { 6 }$
B) $\frac { x ^ { 6 } } { 3 }$
C) $9 x ^ { 6 }$
D) $\frac { 1 } { 3 x ^ { 6 } }$

Question 80

Simplify: $\left( \frac { 81 y ^ { 8 } } { 16 } \right) ^ { - \frac { 3 } { 4 } }$

A) $\frac { 64 } { 729 y ^ { 6 } }$
B) $- \frac { 27 y ^ { 6 } } { 8 }$
C) $\frac { 8 } { 27 y ^ { 6 } }$
D) $\frac { 3 y ^ { 2 } } { 2 }$