Deck 7: Rational Exponents, Radicals, and Complex Numbers

Question 1

Find the square root. Assume that all variables represent nonnegative real numbers.
$$\sqrt { \frac { x ^ { 8 } } { 100 } }$$

A) $\frac { x ^ { 4 } } { 100 }$
B) $\frac { x ^ { 8 } } { 10 }$
C) $\frac { x ^ { 6 } } { 10 }$
D) $\frac { x ^ { 4 } } { 10 }$

Answer 1

D

Question 2

Find the square root. Assume that all variables represent nonnegative real numbers.
$$\sqrt { 49 x ^ { 12 } }$$

A) $49 x ^ { 6 }$
B) $7 x ^ { 6 }$
C) $7 x ^ { 12 }$
D) $7 x ^ { 2 }$

Answer 2

B

Question 3

Find all real square roots of the number.
36

A) 12, -12
B) 7, -7
C) 8, -8
D) 6, -6

Answer 3

6, -6

Question 4

Find the square root. Assume that all variables represent nonnegative real numbers.
$\sqrt { \frac { 441 } { 64 } }$

A) $\frac { 55 } { 8 }$
B) $\frac { 7 } { 3 }$
C) $\frac { 11 } { 4 }$
D) $\frac { 21 } { 8 }$

Question 5

Find the square root. Assume that all variables represent nonnegative real numbers.
$\sqrt { x ^ { 40 } }$

A) $x ^ { 40 }$
B) $x ^ { 20 }$
C) $x ^ { 80 }$
D) $x ^ { 2 }$

Question 6

Use a calculator to approximate the square root to 3 decimal places. Check to see that the approximation is reasonable.
$\sqrt { 78 }$

A) $8.837$
B) $78.000$
C) $8.832$
D) $8.829$

Question 7

Use a calculator to approximate the square root to 3 decimal places. Check to see that the approximation is reasonable.
$\sqrt { 618 }$

A) $618.000$
B) $24.857$
C) $24.865$
D) $24.860$

Question 8

Find the square root. Assume that all variables represent nonnegative real numbers.
$\sqrt { 0.36 }$

A) $0.006$
B) $0.6$
C) 6
D) $0.06$

Question 9

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

A) 81
B) 9
C) 27
D) not a real number

Question 10

Use a calculator to approximate the square root to 3 decimal places. Check to see that the approximation is reasonable.
$\sqrt { 7 }$

A) $2.651$
B) $7.000$
C) $2.643$
D) $2.646$

Question 11

Find the cube root.
$$\sqrt [ 3 ] { - 1000 }$$

A) $31.623$
B) $15.811$
C) $- 10$
D) not a real number

Question 12

Find the square root. Assume that all variables represent nonnegative real numbers.
$\sqrt { - 64 }$

A) $- 32$
B) 8
C) $- 8$
D) not a real number

Question 13

Find the cube root.
$$\sqrt [ 3 ] { 512 x ^ { 6 } }$$

A) $22 x ^ { 2 }$
B) $8 x ^ { 2 }$
C) $64 x ^ { 2 }$
D) not a real number

Question 14

Find the square root. Assume that all variables represent nonnegative real numbers.
$- \sqrt { 81 }$

A) 9
B) $- 9$
C) $- 40$
D) not a real number

Question 15

Find the square root. Assume that all variables represent nonnegative real numbers.
$$\sqrt { \frac { 16 z ^ { 16 } } { 25 } }$$

A) $\frac { 4 z ^ { 14 } } { 5 }$
B) $\frac { 4 z ^ { 8 } } { 5 }$
C) $\frac { 4 z ^ { 8 } } { 25 }$
D) $\frac { 16 z ^ { 8 } } { 25 }$

Question 16

Find all real square roots of the number.
1849

A) 45, -45
B) 43, -43
C) 86, -86
D) 44, -44

Question 17

Find all real square roots of the number.
$\frac { 4 } { 121 }$

A) $\frac { 1 } { 30 } , - \frac { 1 } { 30 }$
B) $\frac { 1 } { 6 } , - \frac { 1 } { 6 }$
C) $\frac { 3 } { 11 } , - \frac { 3 } { 11 }$
D) $\frac { 2 } { 11 } , - \frac { 2 } { 11 }$

Question 18

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

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

Question 19

Find the cube root.
$$\sqrt [ 3 ] { - 125 x ^ { 6 } }$$

A) $11 \mathrm { x } ^ { 2 }$
B) $- 5 x ^ { 2 }$
C) $25 x ^ { 2 }$
D) not a real number

Question 20

Find the square root. Assume that all variables represent nonnegative real numbers.
$\sqrt { 256 }$

A) 128
B) 16
C) 17
D) not a real number

Question 21

Find the root. Assume that all variables represent nonnegative real numbers.
$$\sqrt [ 3 ] { x ^ { 42 } }$$

A) $x ^ { 14 }$
B) $\sqrt [ 3 ] { \mathrm { x } ^ { 42 } }$
C) $x ^ { 39 }$
D) $x ^ { 21 }$

Question 22

Find the cube root.
$$\sqrt [ 3 ] { - 125 x ^ { 36 } y ^ { 12 } }$$

A) $5 x ^ { 12 } y ^ { 4 }$
B) $25 \mathrm { x } ^ { 12 } \mathrm { y } ^ { 4 }$
C) $- 5 x ^ { 18 } y ^ { 6 }$
D) $- 5 x ^ { 12 } y ^ { 4 }$

Question 23

Find the cube root.
$$- \sqrt [ 3 ] { - 64 x ^ { 30 } y ^ { 18 } }$$

A) $4 x ^ { 10 } y ^ { 9 }$
B) $- 4 x ^ { 30 } y ^ { 6 }$
C) $16 x ^ { 10 } y ^ { 6 }$
D) $4 x ^ { 10 } y ^ { 6 }$

Question 24

Find the root. Assume that all variables represent nonnegative real numbers.
$$\sqrt [ 4 ] { y ^ { 36 } }$$

A) $y ^ { 18 }$
B) $\sqrt [ 4 ] { y ^ { 36 } }$
C) $y ^ { 32 }$
D) $\mathrm { y } ^ { 9 }$

Question 25

Find the root. Assume that all variables represent nonnegative real numbers.
$- \sqrt [ 4 ] { 256 }$

A) 16
B) $6.233$
C) $- 4$
D) not real number

Question 26

Find the root. Assume that all variables represent nonnegative real numbers.
$- \sqrt [ 5 ] { 243 }$

A) $- \frac { 243 } { 5 }$
B) 3
C) $- 3$
D) not a real number

Question 27

Find the root. Assume that all variables represent nonnegative real numbers.
$\sqrt [ 11 ] { x ^ { 66 } }$

A) $\frac { x ^ { 6 } } { 11 }$
B) $6 x ^ { 6 }$
C) $x ^ { 6 }$
D) $\frac { 11 } { x ^ { 6 } }$

Question 28

Find the cube root.
$\sqrt [ 3 ] { x ^ { 27 } }$

A) $9 x$
B) $x ^ { 3 }$
C) $x ^ { 9 }$
D) $x ^ { 18 }$

Question 29

Find the root. Assume that all variables represent nonnegative real numbers.
$\sqrt [ 4 ] { 1 }$

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

Question 30

Find the root. Assume that all variables represent nonnegative real numbers.
$- \sqrt [ 4 ] { 256 x ^ { 12 } y ^ { 8 } }$

A) $6.233 x ^ { 3 } y ^ { 2 }$
B) $- 4 x ^ { 3 } y ^ { 2 }$
C) $16 x ^ { 3 } y ^ { 2 }$
D) not a real number

Question 31

Find the root. Assume that all variables represent nonnegative real numbers.
$\sqrt [ 5 ] { 243 }$

A) $\frac { 243 } { 5 }$
B) 4
C) 3
D) not a real number

Question 32

Find the cube root.
$\sqrt [ 3 ] { \frac { x ^ { 12 } } { 8 y ^ { 6 } } }$

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

Question 33

Find the root. Assume that all variables represent nonnegative real numbers.
$\sqrt [ 4 ] { 16 x ^ { 8 } y ^ { 16 } }$

A) $2 x ^ { 2 } y ^ { 4 }$
B) $4 x ^ { 4 } y ^ { 2 }$
C) $2.497 x ^ { 2 } y ^ { 4 }$
D) not a real number

Question 34

Find the root. Assume that all variables represent nonnegative real numbers.
$\sqrt [ 5 ] { \frac { 243 x ^ { 5 } } { y ^ { 20 } } }$

A) $\frac { 3 x } { y ^ { 20 } }$
B) $\frac { 3 x } { y ^ { 4 } }$
C) $\frac { x } { v ^ { 4 } }$
D) $\frac { 3 } { \mathrm { y } ^ { 4 } }$

Question 35

Find the root. Assume that all variables represent nonnegative real numbers.
$\sqrt [ 4 ] { - 16 }$

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

Question 36

Find the root. Assume that all variables represent nonnegative real numbers.
$\sqrt [ 6 ] { 729 }$

A) 3
B) 27
C) 9
D) not a real number

Question 37

Find the root. Assume that all variables represent nonnegative real numbers.
$\sqrt [ 4 ] { \frac { 625 } { x ^ { 12 } } }$

A) $\frac { 625 } { 4 } x ^ { 3 }$
B) $\frac { 5 } { x ^ { 3 } }$
C) $\frac { x ^ { 3 } } { 5 }$
D) $\frac { 5 } { 4 } x ^ { 3 }$

Question 38

Find the root. Assume that all variables represent nonnegative real numbers.
$\sqrt [ 4 ] { 81 }$

A) 3
B) 9
C) $4.264$
D) not real number

Question 39

Simplify. Assume that all variables represent any real number.
$\sqrt [ 3 ] { ( - 11 ) ^ { 3 } }$

A) $- 33$
B) 11
C) $- 11$
D) 33

Question 40

Simplify. Assume that all variables represent any real number.
$$\sqrt { ( - 8 ) ^ { 2 } }$$

A) $- 64$
B) 8
C) $- 8$
D) 64

Question 41

Evaluate.
If $f ( x ) = \sqrt [ 3 ] { x - 1 }$ , find the value of $f ( - 1 )$

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

Question 42

Use radical notation to write the expression. Simplify if possible.
$$( - 32 ) ^ { 1 / 5 }$$

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

Question 43

Simplify. Assume that all variables represent any real number.
$$\sqrt { ( x + 8 ) ^ { 2 } }$$

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

Question 44

Use radical notation to write the expression. Simplify if possible.
$$256 ^ { 1 / 4 }$$

A) 16
B) 64
C) 256
D) 4

Question 45

Evaluate.
If $f ( x ) = \sqrt { 2 x - 7 }$ , find the value of $f ( 11 )$ .

A) $\sqrt { 15 }$
B) 22
C) 15
D) $\sqrt { 22 }$

Question 46

Use radical notation to write the expression. Simplify if possible.
$$- \left( \frac { 81 } { 256 } \right) ^ { 1 / 4 }$$

A) $\frac { 3 } { 5 }$
B) $\frac { 3 } { 4 }$
C) $- \frac { 1 } { 2 }$
D) $- \frac { 3 } { 4 }$

Question 47

Evaluate.
If $f ( x ) = \sqrt { 2 x - 1 }$ , find the value of $f ( 41 )$ .

A) 81
B) 82
C) $\sqrt { 82 }$
D) 9

Question 48

Evaluate.
If $f ( x ) = \sqrt [ 3 ] { x + 20 }$ , find the value of $f ( 7 )$

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

Question 49

Use radical notation to write the expression. Simplify if possible.
$$91 / 2$$

A) 1.5
B) 12
C) 6
D) 3

Question 50

Use radical notation to write the expression. Simplify if possible.
$9 x ^ { 1 / 5 }$

A) $\frac { 9 } { 5 } x$
B) $\sqrt [ 5 ] { 9 \mathrm { x } }$
C) $9 x$
D) $9 \sqrt [ 5 ] { x }$

Question 51

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

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

Question 52

Evaluate.
If $f ( x ) = \sqrt { 2 x + 4 }$ , find the value of $f ( 5 )$ .

A) $\sqrt { 14 }$
B) $\sqrt { 10 }$
C) 10
D) 14

Question 53

Simplify. Assume that all variables represent any real number.
$\sqrt [ 6 ] { x ^ { 6 } }$

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

Question 54

Use radical notation to write the expression. Simplify if possible.
$$\left( \frac { 9 } { 16 } \right) ^ { 1 / 2 }$$

A) $\frac { 2 } { 5 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 3 } { 5 }$
D) $\frac { 1 } { 2 }$

Question 55

Simplify. Assume that all variables represent any real number.
$$\sqrt { x ^ { 2 } + 18 x + 81 }$$

A) $x + 9$
B) $- x - 9$
C) $| x + 9 |$
D) $- | x + 9 |$

Question 56

Use radical notation to write the expression. Simplify if possible.
$$- 36^{1 / 2}$$

A) 12
B) - 18
C) -6
D) 24

Question 57

Use radical notation to write the expression. Simplify if possible.
$$- 243 ^ { 1 / 5 }$$

A) 81
B) 243
C) -3
D) -27

Question 58

Evaluate.
If $f ( x ) = \sqrt { 2 x - 2 }$ , find the value of $f ( 3 )$

A) 4
B) $\sqrt { 6 }$
C) 2
D) 6

Question 59

Simplify. Assume that all variables represent any real number.
$\sqrt [ 5 ] { \mathrm { x } ^ { 5 } }$

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

Question 60

Simplify. Assume that all variables represent any real number.
$\sqrt { 81 x ^ { 2 } }$

A) $9 | \times |$
B) $9 x$
C) $- 9 | x |$
D) $- 9 x$

Question 61

Use radical notation to write the expression. Simplify if possible.
$64 ^ { 4 / 3 }$

A) 16,384
B) 4096
C) 256
D) 1024

Question 62

Use radical notation to write the expression. Simplify if possible.
$7 x ^ { 3 / 10 }$

A) $\sqrt [ 10 ] { 343 x ^ { 3 } }$
B) $\sqrt [ 10 ] { 7 x ^ { 3 } }$
C) $7 \sqrt [ 10 ] { \mathrm { x } ^ { 3 } }$
D) $\sqrt [ 3 ] { 7 x ^ { 10 } }$

Question 63

Write with positive exponents. Simplify if possible.
$$- 8 ^ { - 4 / 3 }$$

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

Question 64

Write with positive exponents. Simplify if possible.
$81 ^ { - 5 / 4 }$

A) 243
B) $\frac { 1 } { 243 }$
C) $- \frac { 1 } { 243 }$
D) not a real number

Question 65

Write with positive exponents. Simplify if possible.
$$\frac { 1 } { x ^ { - 8 / 3 } }$$

A) $\frac { - 1 } { x ^ { 8 / 3 } }$
B) $\frac { 1 } { x ^ { 3 / 8 } }$
C) $- x ^ { 8 / 3 }$
D) $x ^ { 8 / 3 }$

Question 66

Use radical notation to write the expression. Simplify if possible.
$( - 8 ) ^ { 4 / 3 }$

A) $- 16$
B) 16
C) 64
D) not a real number

Question 67

Use radical notation to write the expression. Simplify if possible.
$$\left( - 125 x ^ { 15 } \right) ^ { 1 / 3 }$$

A) $- 5 \sqrt [ 5 ] { x }$
B) $- 5 x ^ { 5 }$
C) $- 5 x ^ { 15 }$
D) $- 125 x ^ { 5 }$

Question 68

Use radical notation to write the expression. Simplify if possible.
$\left( 512 x ^ { 15 } \right) ^ { 1 / 3 }$

A) $512 x ^ { 5 }$
B) $8 x ^ { 5 }$
C) $8 \sqrt [ 5 ] { x }$
D) $8 \mathrm { x } ^ { 15 }$

Question 69

Write with positive exponents. Simplify if possible.
$32 ^ { - 4 / 5 }$

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

Question 70

Write with positive exponents. Simplify if possible.
$$- 243 ^ { - 4 / 5 }$$

A) $\frac { 1 } { 81 }$
B) 81
C) $- \frac { 1 } { 81 }$
D) not a real number

Question 71

Write with positive exponents. Simplify if possible.
$\frac { 7 } { 6 x ^ { - 2 / 7 } }$

A) $42 x ^ { 2 / 7 }$
B) $\frac { 6 x ^ { 2 / 7 } } { 7 }$
C) $\frac { 7 x ^ { 2 / 7 } } { 6 }$
D) $\frac { 36 ^ { 1 / 7 } x ^ { 2 / 7 } } { 7 }$

Question 72

Use radical notation to write the expression. Simplify if possible.
$$( 2 x + 2 ) ^ { 2 / 3 }$$

A) $\sqrt [ 3 ] { 4 x ^ { 2 } + 4 }$
B) $\sqrt [ 2 ] { ( 2 x + 2 ) ^ { 3 } }$
C) $\sqrt [ 3 ] { ( 2 x + 2 ) ^ { 2 } }$
D) $\sqrt [ 3 ] { 2 x ^ { 2 } + 4 }$

Question 73

Use radical notation to write the expression. Simplify if possible.
$\left( \frac { 8 } { 64 } \right) ^ { 2 / 3 }$

A) $- \frac { 1 } { 4 }$
B) $\frac { 1 } { 2 }$
C) $\frac { 1 } { 6 }$
D) $\frac { 1 } { 4 }$

Question 74

Use radical notation to write the expression. Simplify if possible.
$81 ^ { 5 / 4 }$

A) 2187
B) 243
C) 19,683
D) 6561

Question 75

Write with positive exponents. Simplify if possible.
$( - 64 ) ^ { - 2 / 3 }$

A) $- 16$
B) $\frac { 1 } { 16 }$
$C ) - \frac { 1 } { 16 }$
D) $\frac { 128 } { 3 }$

Question 76

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

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

Question 77

Write with positive exponents. Simplify if possible.
$x ^ { - 4 / 9 }$

A) $\frac { 1 } { x ^ { 4 / 9 } }$
B) $\frac { - 1 } { x ^ { 4 / 9 } }$
C) $- x ^ { 4 / 9 }$
D) $\frac { 1 } { x ^ { 9 / 4 } }$

Question 78

Write with positive exponents. Simplify if possible.
$16 ^ { - 3 / 2 }$

A) $- 64$
B) $\frac { 1 } { 64 }$
C) $- \frac { 1 } { 64 }$
D) 64

Question 79

Write with positive exponents. Simplify if possible.
$27 ^ { - 4 / 3 }$

A) $- \frac { 1 } { 81 }$
B) $\frac { 1 } { 81 }$
C) 81
D) not a real number

Question 80

Use radical notation to write the expression. Simplify if possible.
$( 6 x ) ^ { 2 / 5 }$

A) $6 \sqrt [ 5 ] { \mathrm { x } ^ { 2 } }$
B) $\sqrt [ 5 ] { 36 x ^ { 2 } }$
C) $\sqrt [ 5 ] { 6 x ^ { 2 } }$
D) $\sqrt [ 2 ] { 6 x ^ { 5 } }$