Question 1

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

Accepted Answer

A)  $\frac { - 12 - 3 \sqrt { x } } { x + 16 }$ 
B)  $\frac { - 12 + 3 \sqrt { x } } { x ^ { 2 } - 16 }$ 
C)  $\frac { - 12 - 3 \sqrt { x } } { x - 16 }$ 
D)  $\frac { - 12 + 3 \sqrt { x } } { x - 16 }$ 
A)  $\frac { - 12 - 3 \sqrt { x } } { x + 16 }$ 
B)  $\frac { - 12 + 3 \sqrt { x } } { x ^ { 2 } - 16 }$ 
C)  $\frac { - 12 - 3 \sqrt { x } } { x - 16 }$ 
D)  $\frac { - 12 + 3 \sqrt { x } } { x - 16 }$

Question 2

Use the product rule to multiply. Assume all variables represent positive real numbers.
-$\sqrt { 3 x ^ { 3 } } \cdot \sqrt { 3 x ^ { 5 } }$

Accepted Answer

A)  $\sqrt { 3 x ^ { 4 } }$ 
B)  $x 4 \sqrt { 6 }$ 
C)  $3 x ^ { 4 }$ 
D)  $\sqrt { 9 x ^ { 8 } }$ 
A)  $\sqrt { 3 x ^ { 4 } }$ 
B)  $x 4 \sqrt { 6 }$ 
C)  $3 x ^ { 4 }$ 
D)  $\sqrt { 9 x ^ { 8 } }$

Question 3

Simplify. Assume that all variables represent positive real numbers.
-$\sqrt [ 4 ] { x ^ { 27 } }$

Accepted Answer

A)  $x ^ { 6 } \sqrt { 3 }$ 
B)  $x ^ { 4 } \sqrt [ 6 ] { x ^ { 3 } }$ 
C)  $x \sqrt [ 4 ] { x ^ { 3 } }$ 
D)  $x ^ { 4 } \sqrt { 3 }$ 
A)  $x ^ { 6 } \sqrt { 3 }$ 
B)  $x ^ { 4 } \sqrt [ 6 ] { x ^ { 3 } }$ 
C)  $x \sqrt [ 4 ] { x ^ { 3 } }$ 
D)  $x ^ { 4 } \sqrt { 3 }$

Question 4

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

Accepted Answer

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

Question 5

Add or subtract. Assume all variables represent positive real numbers.
-$$3 \sqrt [ 3 ] { x ^ { 3 } y ^ { 13 } } + 4 x y \sqrt [ 3 ] { 8 y ^ { 10 } }$$

Accepted Answer

A)  $11 x y ^ { 4 } \sqrt [ 3 ] { y }$ 
B)  $7 x y ^ { 5 } \sqrt [ 3 ] { 8 y }$ 
C)  $7 x ^ { 2 } y ^ { 5 } \sqrt [ 3 ] { y }$ 
D)  $11 x ^ { 2 } y ^ { 4 } \sqrt [ 3 ] { 2 y }$ 
A)  $11 x y ^ { 4 } \sqrt [ 3 ] { y }$ 
B)  $7 x y ^ { 5 } \sqrt [ 3 ] { 8 y }$ 
C)  $7 x ^ { 2 } y ^ { 5 } \sqrt [ 3 ] { y }$ 
D)  $11 x ^ { 2 } y ^ { 4 } \sqrt [ 3 ] { 2 y }$

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

A)  7 
B)  10 
C)  9 
D)  5 
A)  7 
B)  10 
C)  9 
D)  5

Question 9

Rationalize the numerator and simplify. Assume all variables represent positive real numbers.
-$\frac { \sqrt { x } - 4 \sqrt { y } } { 2 \sqrt { x } }$

Accepted Answer

A)  $\frac { x - 4 y } { 2 x + 8 \sqrt { x y } }$ 
B)  $\frac { x - 16 y } { 2 x + 8 \sqrt { x y } }$ 
C)  $\frac { x + 16 y } { 2 x - 8 \sqrt { x y } }$ 
D)  $\frac { x - 16 y } { 2 x - 8 \sqrt { x y } }$ 
A)  $\frac { x - 4 y } { 2 x + 8 \sqrt { x y } }$ 
B)  $\frac { x - 16 y } { 2 x + 8 \sqrt { x y } }$ 
C)  $\frac { x + 16 y } { 2 x - 8 \sqrt { x y } }$ 
D)  $\frac { x - 16 y } { 2 x - 8 \sqrt { x y } }$

Question 10

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

Accepted Answer

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

Question 11

Solve.
-$$\sqrt [ 3 ] { 2 x } = - 4$$

Accepted Answer

A)  32 
B)  $- 32$ 
C)  8 
D)  $\varnothing$ 
A)  32 
B)  $- 32$ 
C)  8 
D)  $\varnothing$

Question 12

Rationalize the denominator. Assume that all variables represent positive real numbers.
-$\frac { \sqrt { 13 } + \sqrt { 2 } } { \sqrt { 13 } - \sqrt { 2 } }$

Accepted Answer

A)  $\frac { 173 + 2 \sqrt { 26 } } { 165 }$ 
B)  $\frac { 15 + 2 \sqrt { 26 } } { 11 }$ 
C)  $\frac { 173 + 2 \sqrt { 26 } } { 11 }$ 
D)  $\frac { 15 + 2 \sqrt { 26 } } { 165 }$ 
A)  $\frac { 173 + 2 \sqrt { 26 } } { 165 }$ 
B)  $\frac { 15 + 2 \sqrt { 26 } } { 11 }$ 
C)  $\frac { 173 + 2 \sqrt { 26 } } { 11 }$ 
D)  $\frac { 15 + 2 \sqrt { 26 } } { 165 }$

Question 13

Solve.
-$$\sqrt { 3 x + 2 } + 7 = 14$$

Accepted Answer

A)  $\frac { 47 } { 3 }$ 
B)  141 
C)  $\frac { 3 } { 47 }$ 
D)  $\varnothing$ 
A)  $\frac { 47 } { 3 }$ 
B)  141 
C)  $\frac { 3 } { 47 }$ 
D)  $\varnothing$

Question 14

Solve.
-$$\sqrt [ 3 ] { 4 x + 3 } + 2 = 0$$

Accepted Answer

A)  $\frac { 1 } { 4 }$ 
B)  $- \frac { 11 } { 4 }$ 
C)  $- 3$ 
D)  $- 2$ 
A)  $\frac { 1 } { 4 }$ 
B)  $- \frac { 11 } { 4 }$ 
C)  $- 3$ 
D)  $- 2$

Question 15

Solve.
-$$\sqrt { x + 58 } - 2 = \sqrt { x + 34 }$$

Accepted Answer

A)  10 
B)  9 
C)  $- 9$ 
D)  $\varnothing$ 
A)  10 
B)  9 
C)  $- 9$ 
D)  $\varnothing$

Question 16

Perform the indicated operation. Write the result in the form a + bi.
-$$\frac { 8 + 4 i } { 8 + 9 i }$$

Accepted Answer

A)  $\mathrm {  }  - \frac { 20 } { 17 } + \frac { 8 } { 17 } \mathrm { i }$ 
B)  $- \frac { 28 } { 17 } + \frac { 8 } { 17 } i$ 
C)  $\frac { 28 } { 29 } - \frac { 104 } { 29 } \mathrm { i }$ 
D)  $\frac { 20 } { 29 } - \frac { 8 } { 29 } i$ 
A)  $\mathrm {  }  - \frac { 20 } { 17 } + \frac { 8 } { 17 } \mathrm { i }$ 
B)  $- \frac { 28 } { 17 } + \frac { 8 } { 17 } i$ 
C)  $\frac { 28 } { 29 } - \frac { 104 } { 29 } \mathrm { i }$ 
D)  $\frac { 20 } { 29 } - \frac { 8 } { 29 } i$

Question 17

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

Accepted Answer

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

Question 18

Add or subtract. Assume all variables represent positive real numbers.
-$$12 \sqrt [ 3 ] { 3 } + 9 \sqrt [ 3 ] { 3 }$$

Accepted Answer

A)  $21 \sqrt [ 3 ] { 6 }$ 
B)  $- 3 \sqrt [ 3 ] { 3 }$ 
C)  $21 \sqrt [ 3 ] { 3 }$ 
D)  $21 \sqrt [ 3 ] { 9 }$ 
A)  $21 \sqrt [ 3 ] { 6 }$ 
B)  $- 3 \sqrt [ 3 ] { 3 }$ 
C)  $21 \sqrt [ 3 ] { 3 }$ 
D)  $21 \sqrt [ 3 ] { 9 }$

Question 19

Multiply, and then simplify if possible. Assume all variables represent positive real numbers.
-$( \sqrt { m } - y ) ( \sqrt { m } + y )$

Accepted Answer

A)  $\mathrm { m } - \mathrm { y }$ 
B)  $m - y ^ { 2 }$ 
C)  $m - 2 y \sqrt { m } - y ^ { 2 }$ 
D)  $m + 2 y \sqrt { m } - y ^ { 2 }$ 
A)  $\mathrm { m } - \mathrm { y }$ 
B)  $m - y ^ { 2 }$ 
C)  $m - 2 y \sqrt { m } - y ^ { 2 }$ 
D)  $m + 2 y \sqrt { m } - y ^ { 2 }$

Question 20

Provide an appropriate response.
-If $g ( x ) = \sqrt { 2 x + 3 }$, find $g ( 0 )$ and $g ( 39 )$.

Accepted Answer

A)  $\mathrm { g } ( 0 ) = 0 ; \mathrm { g } ( 39 ) = 81$ 
B)  $\mathrm { g } ( 0 ) = 0 ; \mathrm { g } ( 39 ) = 9$ 
C)  $g ( 0 ) = \sqrt { 3 } ; g ( 39 ) = 81$ 
D)  $g ( 0 ) = \sqrt { 3 } ; g ( 39 ) = 9$ 
A)  $\mathrm { g } ( 0 ) = 0 ; \mathrm { g } ( 39 ) = 81$ 
B)  $\mathrm { g } ( 0 ) = 0 ; \mathrm { g } ( 39 ) = 9$ 
C)  $g ( 0 ) = \sqrt { 3 } ; g ( 39 ) = 81$ 
D)  $g ( 0 ) = \sqrt { 3 } ; g ( 39 ) = 9$

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

Use the product rule to multiply. Assume all variables represent positive real numbers. - $\sqrt { 3 x ^ { 3 } } \cdot \sqrt { 3 x ^ { 5 } }$

Simplify. Assume that all variables represent positive real numbers. - $\sqrt [ 4 ] { x ^ { 27 } }$

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

Add or subtract. Assume all variables represent positive real numbers. - $3 \sqrt [ 3 ] { x ^ { 3 } y ^ { 13 } } + 4 x y \sqrt [ 3 ] { 8 y ^ { 10 } }$

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

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

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

Rationalize the numerator and simplify. Assume all variables represent positive real numbers. - $\frac { \sqrt { x } - 4 \sqrt { y } } { 2 \sqrt { x } }$

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

Solve. - $\sqrt [ 3 ] { 2 x } = - 4$

Rationalize the denominator. Assume that all variables represent positive real numbers. - $\frac { \sqrt { 13 } + \sqrt { 2 } } { \sqrt { 13 } - \sqrt { 2 } }$

Solve. - $\sqrt { 3 x + 2 } + 7 = 14$

Solve. - $\sqrt [ 3 ] { 4 x + 3 } + 2 = 0$

Solve. - $\sqrt { x + 58 } - 2 = \sqrt { x + 34 }$

Perform the indicated operation. Write the result in the form a + bi. - $\frac { 8 + 4 i } { 8 + 9 i }$

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

Add or subtract. Assume all variables represent positive real numbers. - $12 \sqrt [ 3 ] { 3 } + 9 \sqrt [ 3 ] { 3 }$

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - $( \sqrt { m } - y ) ( \sqrt { m } + y )$

Provide an appropriate response. -If $g ( x ) = \sqrt { 2 x + 3 }$ , find $g ( 0 )$ and $g ( 39 )$ .

Real Numbers and Algebraic Expressions

Equations, Inequalities, and Problem Solving

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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. - 3x+4\frac { 3 } { \sqrt { x } + 4 }x​+43​

Use the product rule to multiply. Assume all variables represent positive real numbers. - 3x3⋅3x5\sqrt { 3 x ^ { 3 } } \cdot \sqrt { 3 x ^ { 5 } }3x3​⋅3x5​

Simplify. Assume that all variables represent positive real numbers. - x274\sqrt [ 4 ] { x ^ { 27 } }4x27​

Find the root. Assume that all variables represent nonnegative real numbers. - y364\sqrt [ 4 ] { y ^ { 36 } }4y36​

Add or subtract. Assume all variables represent positive real numbers. - 3x3y133+4xy8y1033 \sqrt [ 3 ] { x ^ { 3 } y ^ { 13 } } + 4 x y \sqrt [ 3 ] { 8 y ^ { 10 } }33x3y13​+4xy38y10​

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

Find the cube root. - x273\sqrt [ 3 ] { x ^ { 27 } }3x27​