Question 1

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

Accepted Answer

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

Question 2

Fill in the blank with one of the words or phrases listed below.

-The process of writing a radical expression as an equivalent expression but without a radical in the denominator is called ــــــــــــــ the denominator.

Accepted Answer

A)  rationalizing 
B)  radicand 
C)  index 
D)  imaginary unit 
A)  rationalizing 
B)  radicand 
C)  index 
D)  imaginary unit

Question 3

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

Accepted Answer

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

Question 4

Solve the problem.
-The maximum distance $\mathrm { d }$ in kilometers that you can see from a height $h$ in meters is given by the formula $d = 3.5 \sqrt { h }$. How high above the ground must you be to see 40 kilometers? Round to the nearest tenth of a meter.

Accepted Answer

A)  $457.1 \mathrm {~m}$ 
B)  $130.6 \mathrm {~m}$ 
C)  $11.4 \mathrm {~m}$ 
D)  $22.1 \mathrm {~m}$ 
A)  $457.1 \mathrm {~m}$ 
B)  $130.6 \mathrm {~m}$ 
C)  $11.4 \mathrm {~m}$ 
D)  $22.1 \mathrm {~m}$

Question 5

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

Accepted Answer

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

Question 6

Find the power of i.
-$\mathrm { i } ^ { 58 }$

Accepted Answer

A)  $- 1$ 
B)  $- \mathrm { i }$ 
C)  $\mathrm { i }$ 
D)  1 
A)  $- 1$ 
B)  $- \mathrm { i }$ 
C)  $\mathrm { i }$ 
D)  1

Question 7

Perform the indicated operation. Write the result in the form a + bi.
-$\frac { 1 - 21 i } { - 7 i }$

Accepted Answer

A)  $- \frac { 1 } { 7 } + 3 i$ 
B)  $3 + \frac { 1 } { 7 } \mathrm { i }$ 
C)  $\frac { 1 } { 7 } - 3 i$ 
D)  $- 3 - \frac { 1 } { 7 } \mathrm { i }$ 
A)  $- \frac { 1 } { 7 } + 3 i$ 
B)  $3 + \frac { 1 } { 7 } \mathrm { i }$ 
C)  $\frac { 1 } { 7 } - 3 i$ 
D)  $- 3 - \frac { 1 } { 7 } \mathrm { i }$

Question 8

Use rational exponents to write as a single radical expression.
-$\sqrt [ 3 ] { 5 } \cdot \sqrt { 3 }$

Accepted Answer

A)  $\sqrt [ 6 ] { 135 }$ 
B)  $\sqrt [ 6 ] { 675 }$ 
C)  $\sqrt [ 3 ] { 15 }$ 
D)  $\sqrt [ 6 ] { 15 }$ 
A)  $\sqrt [ 6 ] { 135 }$ 
B)  $\sqrt [ 6 ] { 675 }$ 
C)  $\sqrt [ 3 ] { 15 }$ 
D)  $\sqrt [ 6 ] { 15 }$

Question 9

Add or subtract. Assume all variables represent positive real numbers.
-$$5 \sqrt [ 3 ] { a } + \sqrt [ 3 ] { 64 a }$$

Accepted Answer

A)  $20 \sqrt [ 3 ] { a }$ 
B)  $9 \sqrt [ 3 ] { a }$ 
C)  $5 \sqrt [ 3 ] { a } + \sqrt [ 3 ] { 64 a }$ 
D)  $6 \sqrt [ 3 ] { 64 a }$ 
A)  $20 \sqrt [ 3 ] { a }$ 
B)  $9 \sqrt [ 3 ] { a }$ 
C)  $5 \sqrt [ 3 ] { a } + \sqrt [ 3 ] { 64 a }$ 
D)  $6 \sqrt [ 3 ] { 64 a }$

Question 10

Use the Pythagorean theorem to find the unknown side of the right triangle.
-

Accepted Answer

A)  4 
B)  22 
C)  $\sqrt { 22 }$ 
D)  $2 \sqrt { 7 }$ 
A)  4 
B)  22 
C)  $\sqrt { 22 }$ 
D)  $2 \sqrt { 7 }$

Question 11

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

Accepted Answer

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

Question 12

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\sqrt [ 3 ] { \frac { 4 } { 9 } }$

Accepted Answer

A)  $\frac { \sqrt [ 3 ] { 324 } } { 9 }$ 
B)  $\frac { \sqrt [ 3 ] { 12 } } { 3 }$ 
C)  $\frac { \sqrt [ 3 ] { 36 } } { 9 }$ 
D)  $\frac { \sqrt [ 3 ] { 324 } } { 81 }$ 
A)  $\frac { \sqrt [ 3 ] { 324 } } { 9 }$ 
B)  $\frac { \sqrt [ 3 ] { 12 } } { 3 }$ 
C)  $\frac { \sqrt [ 3 ] { 36 } } { 9 }$ 
D)  $\frac { \sqrt [ 3 ] { 324 } } { 81 }$

Question 13

Simplify. Assume that all variables represent positive real numbers.
-$\frac { 5 \sqrt { 8 x ^ { 3 } } } { 11 \sqrt { 2 x ^ { - 6 } } }$

Accepted Answer

A)  $\frac { 30 x ^ { 5 } } { 11 }$ 
B)  $20 x ^ { 4 } \sqrt { x }$ 
C)  $\frac { 10 x ^ { 4 } \sqrt { x } } { 11 }$ 
D)  $\frac { 20 x ^ { 3 } \sqrt { x ^ { 3 } } } { 11 }$ 
A)  $\frac { 30 x ^ { 5 } } { 11 }$ 
B)  $20 x ^ { 4 } \sqrt { x }$ 
C)  $\frac { 10 x ^ { 4 } \sqrt { x } } { 11 }$ 
D)  $\frac { 20 x ^ { 3 } \sqrt { x ^ { 3 } } } { 11 }$

Question 14

Use the properties of exponents to simplify. Write with positive exponents.
-$\frac { x ^ { 3 / 5 } \cdot x ^ { 6 / 7 } } { x ^ { - 1 / 3 } }$

Accepted Answer

A)  $\frac { 1 } { \mathrm { x } ^ { 188 / 105 } }$ 
B)  $x ^ { 188 / 105 }$ 
C)  $\frac { 1 } { x ^ { 118 / 105 } }$ 
D)  $x ^ { 118 / 105 }$ 
A)  $\frac { 1 } { \mathrm { x } ^ { 188 / 105 } }$ 
B)  $x ^ { 188 / 105 }$ 
C)  $\frac { 1 } { x ^ { 118 / 105 } }$ 
D)  $x ^ { 118 / 105 }$

Question 15

Write in terms of i.
-$\sqrt { - 168 }$

Accepted Answer

A)  $2 \mathrm { i } \sqrt { 42 }$ 
B)  $- 2 \sqrt { 42 }$ 
C)  $- 2 \mathrm { i } \sqrt { 42 }$ 
D)  $2 \sqrt { 42 }$ 
A)  $2 \mathrm { i } \sqrt { 42 }$ 
B)  $- 2 \sqrt { 42 }$ 
C)  $- 2 \mathrm { i } \sqrt { 42 }$ 
D)  $2 \sqrt { 42 }$

Question 16

Solve.
-$$\sqrt { 3 x - 6 } = \sqrt { 2 x + 6 }$$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ 0 \}$ 
C)  $\{ 12 \}$ 
D)  $\{ - 12 \}$ 
A)  $\varnothing$ 
B)  $\{ 0 \}$ 
C)  $\{ 12 \}$ 
D)  $\{ - 12 \}$

Question 17

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

Accepted Answer

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

Question 18

Use the product rule to multiply. Assume all variables represent positive real numbers.
-$\sqrt { 150 } \cdot \sqrt { 6 }$

Accepted Answer

A)  30 
B)  $6 \sqrt { 5 }$ 
C)  900 
D)  $5 \sqrt { 6 }$ 
A)  30 
B)  $6 \sqrt { 5 }$ 
C)  900 
D)  $5 \sqrt { 6 }$

Question 19

Rationalize the numerator and simplify. Assume all variables represent positive real numbers.
-$\frac { 7 \sqrt { x } } { \sqrt { 2 y } }$

Accepted Answer

A)  $\frac { 7 x } { \sqrt { 2 x y } }$ 
B)  $\frac { 7 \sqrt { 2 x y } } { 2 y }$ 
C)  $\frac { 7 x } { \sqrt { 2 y } }$ 
D)  $\frac { 7 x } { \sqrt { 14 x y } }$ 
A)  $\frac { 7 x } { \sqrt { 2 x y } }$ 
B)  $\frac { 7 \sqrt { 2 x y } } { 2 y }$ 
C)  $\frac { 7 x } { \sqrt { 2 y } }$ 
D)  $\frac { 7 x } { \sqrt { 14 x y } }$

Question 20

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

Accepted Answer

A)  $b ^ { 2 / 25 }$ 
B)  $b ^ { 7 / 5 }$ 
C)  $b ^ { 2 }$ 
D)  $b ^ { 1 / 2 }$ 
A)  $b ^ { 2 / 25 }$ 
B)  $b ^ { 7 / 5 }$ 
C)  $b ^ { 2 }$ 
D)  $b ^ { 1 / 2 }$

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

Fill in the blank with one of the words or phrases listed below. -The process of writing a radical expression as an equivalent expression but without a radical in the denominator is called ــــــــــــــ the denominator.

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

Solve the problem. -The maximum distance $\mathrm { d }$ in kilometers that you can see from a height $h$ in meters is given by the formula $d = 3.5 \sqrt { h }$ . How high above the ground must you be to see 40 kilometers? Round to the nearest tenth of a meter.

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

Find the power of i. - $\mathrm { i } ^ { 58 }$

Perform the indicated operation. Write the result in the form a + bi. - $\frac { 1 - 21 i } { - 7 i }$

Use rational exponents to write as a single radical expression. - $\sqrt [ 3 ] { 5 } \cdot \sqrt { 3 }$

Add or subtract. Assume all variables represent positive real numbers. - $5 \sqrt [ 3 ] { a } + \sqrt [ 3 ] { 64 a }$

Use the Pythagorean theorem to find the unknown side of the right triangle. -

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

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\sqrt [ 3 ] { \frac { 4 } { 9 } }$

Simplify. Assume that all variables represent positive real numbers. - $\frac { 5 \sqrt { 8 x ^ { 3 } } } { 11 \sqrt { 2 x ^ { - 6 } } }$

Use the properties of exponents to simplify. Write with positive exponents. - $\frac { x ^ { 3 / 5 } \cdot x ^ { 6 / 7 } } { x ^ { - 1 / 3 } }$

Write in terms of i. - $\sqrt { - 168 }$

Solve. - $\sqrt { 3 x - 6 } = \sqrt { 2 x + 6 }$

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

Use the product rule to multiply. Assume all variables represent positive real numbers. - $\sqrt { 150 } \cdot \sqrt { 6 }$

Rationalize the numerator and simplify. Assume all variables represent positive real numbers. - $\frac { 7 \sqrt { x } } { \sqrt { 2 y } }$

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

Real Numbers and Algebraic Expressions

Equations, Inequalities, and Problem Solving

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Exam 7: Rational Exponents, Radicals, and Complex Numbers

Find the root. Assume that all variables represent nonnegative real numbers. - 243x5y205\sqrt [ 5 ] { \frac { 243 x ^ { 5 } } { y ^ { 20 } } }5y20243x5​​

Fill in the blank with one of the words or phrases listed below. -The process of writing a radical expression as an equivalent expression but without a radical in the denominator is called ــــــــــــــ the denominator.

Find the root. Assume that all variables represent nonnegative real numbers. - 814\sqrt [ 4 ] { 81 }481​

Solve the problem. -The maximum distance d\mathrm { d }d in kilometers that you can see from a height hhh in meters is given by the formula d=3.5hd = 3.5 \sqrt { h }d=3.5h​ . How high above the ground must you be to see 40 kilometers? Round to the nearest tenth of a meter.

Write with positive exponents. Simplify if possible. - 32−4/532 ^ { - 4 / 5 }32−4/5

Find the power of i. - i58\mathrm { i } ^ { 58 }i58

Perform the indicated operation. Write the result in the form a + bi. - 1−21i−7i\frac { 1 - 21 i } { - 7 i }−7i1−21i​

Use rational exponents to write as a single radical expression. - 53⋅3\sqrt [ 3 ] { 5 } \cdot \sqrt { 3 }35​⋅3​

Add or subtract. Assume all variables represent positive real numbers. - 5a3+64a35 \sqrt [ 3 ] { a } + \sqrt [ 3 ] { 64 a }53a​+364a​