Question 1

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$$\frac { 2 } { 8 - \sqrt { 5 } }$$

Accepted Answer

A)  $\frac { 2 } { 8 } - \frac { 2 } { \sqrt { 5 } }$ 
B)  $\frac { 16 - 2 \sqrt { 5 } } { 59 }$ 
C)  $\frac { 16 + 2 \sqrt { 5 } } { - 3 }$ 
D)  $\frac { 16 + 2 \sqrt { 5 } } { 59 }$ 
A)  $\frac { 2 } { 8 } - \frac { 2 } { \sqrt { 5 } }$ 
B)  $\frac { 16 - 2 \sqrt { 5 } } { 59 }$ 
C)  $\frac { 16 + 2 \sqrt { 5 } } { - 3 }$ 
D)  $\frac { 16 + 2 \sqrt { 5 } } { 59 }$

Question 2

Perform the indicated operation. Assume that all variables represent positive real numbers.
-$( \sqrt { 7 } + 5 ) ( \sqrt { 7 } - 5 )$

Accepted Answer

A)  $7 - 2 \sqrt { 5 }$ 
B)  $- 18$ 
C)  2 
D)  32 
A)  $7 - 2 \sqrt { 5 }$ 
B)  $- 18$ 
C)  2 
D)  32

Question 3

Simplify. Assume that all variables represent positive real numbers.
-$$\frac { \sqrt [ 3 ] { 54 x ^ { 7 } } } { \sqrt [ 3 ] { 2 x } }$$

Accepted Answer

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

Question 4

Perform the indicated operation. Write the result in the form a + bi.
-$$( 9 - 2 i ) + ( 3 + 5 i )$$

Accepted Answer

A)  $- 12 - 3 \mathrm { i }$ 
B)  $12 - 3 \mathrm { i }$ 
C)  $6 + 7 i$ 
D)  $12 + 3 i$ 
A)  $- 12 - 3 \mathrm { i }$ 
B)  $12 - 3 \mathrm { i }$ 
C)  $6 + 7 i$ 
D)  $12 + 3 i$

Question 5

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

Accepted Answer

A)  $\frac { 4 z ^ { 14 } } { 5 }$ 
B)  $\frac { 4 z ^ { 8 } } { 5 }$ 
C)  $\frac { 4 z ^ { 8 } } { 25 }$ 
D)  $\frac { 16 z ^ { 8 } } { 25 }$ 
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 6

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\frac { 7 } { \sqrt [ 4 ] { a ^ { 3 } } }$

Accepted Answer

A)  $\frac { 7 \sqrt [ 4 ] { a } } { a }$ 
B)  $7 \sqrt [ 4 ] { a }$ 
C)  $\frac { 7 \sqrt [ 4 ] { a ^ { 3 } } } { a }$ 
D)  $\frac { 7 \sqrt [ 4 ] { a ^ { 2 } } } { a }$ 
A)  $\frac { 7 \sqrt [ 4 ] { a } } { a }$ 
B)  $7 \sqrt [ 4 ] { a }$ 
C)  $\frac { 7 \sqrt [ 4 ] { a ^ { 3 } } } { a }$ 
D)  $\frac { 7 \sqrt [ 4 ] { a ^ { 2 } } } { a }$

Question 7

Perform the indicated operation. Assume that all variables represent positive real numbers.
-$\sqrt { 7 } ( \sqrt { 5 } - \sqrt { 3 } )$

Accepted Answer

A)  $\sqrt { 35 } - \sqrt { 21 }$ 
B)  $7 \sqrt { 5 } + 7 \sqrt { 3 }$ 
C)  $\sqrt { 56 }$ 
D)  $8 \sqrt { 7 }$ 
A)  $\sqrt { 35 } - \sqrt { 21 }$ 
B)  $7 \sqrt { 5 } + 7 \sqrt { 3 }$ 
C)  $\sqrt { 56 }$ 
D)  $8 \sqrt { 7 }$

Question 8

Use rational exponents to simplify. Assume that all variables represent positive real numbers.
-$\sqrt [ 20 ] { ( 10 y ) ^ { 4 } }$

Accepted Answer

A)  $\sqrt [ 5 ] { 10 y }$ 
B)  $\sqrt [ 16 ] { 10 y }$ 
C)  $\sqrt [ 5 ] { 10 \mathrm { y } ^ { 4 } }$ 
D)  $100,000 y ^ { 5 }$ 
A)  $\sqrt [ 5 ] { 10 y }$ 
B)  $\sqrt [ 16 ] { 10 y }$ 
C)  $\sqrt [ 5 ] { 10 \mathrm { y } ^ { 4 } }$ 
D)  $100,000 y ^ { 5 }$

Question 9

Find the power of i.
-$i ^ { 43 }$

Accepted Answer

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

Question 10

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

Accepted Answer

A)  185 
B)  $\sqrt { 370 }$ 
C)  11 
D)  370 
A)  185 
B)  $\sqrt { 370 }$ 
C)  11 
D)  370

Question 11

Solve.
-$$\sqrt { x + 2 } - 7 = 0$$

Accepted Answer

A)  81 
B)  49 
C)  51 
D)  47 
A)  81 
B)  49 
C)  51 
D)  47

Question 12

Rationalize the denominator. Assume that all variables represent positive real numbers.
-$\frac { 5 } { \sqrt [ 3 ] { 36 x ^ { 2 } } }$

Accepted Answer

A)  $\frac { 5 \sqrt [ 3 ] { 36 x ^ { 2 } } } { 36 x ^ { 2 } }$ 
B)  $\frac { 5 \sqrt [ 3 ] { x } } { x }$ 
C)  $\frac { 5 \sqrt [ 3 ] { 6 x } } { 6 x }$ 
D)  $\frac { 5 \sqrt { 6 x } } { 6 x }$ 
A)  $\frac { 5 \sqrt [ 3 ] { 36 x ^ { 2 } } } { 36 x ^ { 2 } }$ 
B)  $\frac { 5 \sqrt [ 3 ] { x } } { x }$ 
C)  $\frac { 5 \sqrt [ 3 ] { 6 x } } { 6 x }$ 
D)  $\frac { 5 \sqrt { 6 x } } { 6 x }$

Question 13

Solve.
-$$\sqrt { 20 x - 40 } = x + 3$$

Accepted Answer

A)  9 
B)  $- 6$ 
C)  7 
D)  $- 7$ 
A)  9 
B)  $- 6$ 
C)  7 
D)  $- 7$

Question 14

Find the midpoint of the line segment whose endpoints are given.
-$$\left( \frac { 7 } { 5 } , \frac { 3 } { 2 } \right) , \left( \frac { 4 } { 5 } , \frac { 5 } { 2 } \right)$$

Accepted Answer

A)  $\left( \frac { 11 } { 5 } , 4 \right)$ 
B)  $\left( - \frac { 3 } { 10 } , \frac { 1 } { 2 } \right)$ 
C)  $\left( \frac { 11 } { 10 } , 2 \right)$ 
D)  $\left( \frac { 3 } { 10 } , - \frac { 1 } { 2 } \right)$ 
A)  $\left( \frac { 11 } { 5 } , 4 \right)$ 
B)  $\left( - \frac { 3 } { 10 } , \frac { 1 } { 2 } \right)$ 
C)  $\left( \frac { 11 } { 10 } , 2 \right)$ 
D)  $\left( \frac { 3 } { 10 } , - \frac { 1 } { 2 } \right)$

Question 15

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\frac { \sqrt { 7 } - \sqrt { 8 } } { \sqrt { 7 } + \sqrt { 8 } }$

Accepted Answer

A)  $15 + 2 \sqrt { 56 }$ 
B)  $2 \sqrt { 56 } - 15$ 
C)  $15 - 2 \sqrt { 56 }$ 
D)  $- 15 - 2 \sqrt { 56 }$ 
A)  $15 + 2 \sqrt { 56 }$ 
B)  $2 \sqrt { 56 } - 15$ 
C)  $15 - 2 \sqrt { 56 }$ 
D)  $- 15 - 2 \sqrt { 56 }$

Question 16

Perform the indicated operation. Write the result in the form a + bi.
-$$( 6 + 5 i ) ( 6 - 5 i )$$

Accepted Answer

A)  11 
B)  $36 - 25 \mathrm { i } ^ { 2 }$ 
C)  $36 - 25 \mathrm { i }$ 
D)  61 
A)  11 
B)  $36 - 25 \mathrm { i } ^ { 2 }$ 
C)  $36 - 25 \mathrm { i }$ 
D)  61

Question 17

Rationalize the denominator. Assume that all variables represent positive real numbers.
-$\sqrt { \frac { 25 } { x } }$

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\sqrt { \frac { 1 } { 2 } }$

Accepted Answer

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

Question 20

Simplify. Assume that all variables represent positive real numbers.
-$\sqrt [ 5 ] { 96 }$

Accepted Answer

A)  $2 \sqrt { 3 }$ 
B)  $2 \sqrt [ 5 ] { 3 }$ 
C)  $3 \sqrt [ 5 ] { 2 }$ 
D)  $3 \sqrt { 2 }$ 
A)  $2 \sqrt { 3 }$ 
B)  $2 \sqrt [ 5 ] { 3 }$ 
C)  $3 \sqrt [ 5 ] { 2 }$ 
D)  $3 \sqrt { 2 }$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { 2 } { 8 - \sqrt { 5 } }$

Perform the indicated operation. Assume that all variables represent positive real numbers. - $( \sqrt { 7 } + 5 ) ( \sqrt { 7 } - 5 )$

Simplify. Assume that all variables represent positive real numbers. - $\frac { \sqrt [ 3 ] { 54 x ^ { 7 } } } { \sqrt [ 3 ] { 2 x } }$

Perform the indicated operation. Write the result in the form a + bi. - $( 9 - 2 i ) + ( 3 + 5 i )$

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

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { 7 } { \sqrt [ 4 ] { a ^ { 3 } } }$

Perform the indicated operation. Assume that all variables represent positive real numbers. - $\sqrt { 7 } ( \sqrt { 5 } - \sqrt { 3 } )$

Use rational exponents to simplify. Assume that all variables represent positive real numbers. - $\sqrt [ 20 ] { ( 10 y ) ^ { 4 } }$

Find the power of i. - $i ^ { 43 }$

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

Solve. - $\sqrt { x + 2 } - 7 = 0$

Rationalize the denominator. Assume that all variables represent positive real numbers. - $\frac { 5 } { \sqrt [ 3 ] { 36 x ^ { 2 } } }$

Solve. - $\sqrt { 20 x - 40 } = x + 3$

Find the midpoint of the line segment whose endpoints are given. - $\left( \frac { 7 } { 5 } , \frac { 3 } { 2 } \right) , \left( \frac { 4 } { 5 } , \frac { 5 } { 2 } \right)$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { \sqrt { 7 } - \sqrt { 8 } } { \sqrt { 7 } + \sqrt { 8 } }$

Perform the indicated operation. Write the result in the form a + bi. - $( 6 + 5 i ) ( 6 - 5 i )$

Rationalize the denominator. Assume that all variables represent positive real numbers. - $\sqrt { \frac { 25 } { x } }$

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

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\sqrt { \frac { 1 } { 2 } }$

Simplify. Assume that all variables represent positive real numbers. - $\sqrt [ 5 ] { 96 }$

Real Numbers and Algebraic Expressions

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Exam 7: Rational Exponents, Radicals, and Complex Numbers

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - 28−5\frac { 2 } { 8 - \sqrt { 5 } }8−5​2​

Perform the indicated operation. Assume that all variables represent positive real numbers. - (7+5)(7−5)( \sqrt { 7 } + 5 ) ( \sqrt { 7 } - 5 )(7​+5)(7​−5)

Simplify. Assume that all variables represent positive real numbers. - 54x732x3\frac { \sqrt [ 3 ] { 54 x ^ { 7 } } } { \sqrt [ 3 ] { 2 x } }32x​354x7​​

Perform the indicated operation. Write the result in the form a + bi. - (9−2i)+(3+5i)( 9 - 2 i ) + ( 3 + 5 i )(9−2i)+(3+5i)

Find the square root. Assume that all variables represent nonnegative real numbers. - 16z1625\sqrt { \frac { 16 z ^ { 16 } } { 25 } }2516z16​​

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - 7a34\frac { 7 } { \sqrt [ 4 ] { a ^ { 3 } } }4a3​7​

Perform the indicated operation. Assume that all variables represent positive real numbers. - 7(5−3)\sqrt { 7 } ( \sqrt { 5 } - \sqrt { 3 } )7​(5​−3​)

Use rational exponents to simplify. Assume that all variables represent positive real numbers. - (10y)420\sqrt [ 20 ] { ( 10 y ) ^ { 4 } }20(10y)4​

Find the power of i. - i43i ^ { 43 }i43