Question 1

Multiply, and then simplify if possible. Assume all variables represent positive real numbers.
-$( 6 \sqrt { 5 } + 3 ) ( 6 \sqrt { 5 } + 8 )$

Accepted Answer

A)  $60 \sqrt { 5 } + 48$ 
B)  $204 + 66 \sqrt { 5 }$ 
C)  $108 \sqrt { 5 }$ 
D)  $24 + 36 \sqrt { 5 ^ { 2 } } + 48 \sqrt { 5 }$ 
A)  $60 \sqrt { 5 } + 48$ 
B)  $204 + 66 \sqrt { 5 }$ 
C)  $108 \sqrt { 5 }$ 
D)  $24 + 36 \sqrt { 5 ^ { 2 } } + 48 \sqrt { 5 }$

Question 2

Use rational exponents to write as a single radical expression.
-$\sqrt [ 5 ] { 4 } \cdot \sqrt { 2 }$

Accepted Answer

A)  $\sqrt [ 10 ] { 128 }$ 
B)  $\sqrt [ 5 ] { 8 }$ 
C)  $\sqrt [ 10 ] { 8 }$ 
D)  $\sqrt [ 10 ] { 512 }$ 
A)  $\sqrt [ 10 ] { 128 }$ 
B)  $\sqrt [ 5 ] { 8 }$ 
C)  $\sqrt [ 10 ] { 8 }$ 
D)  $\sqrt [ 10 ] { 512 }$

Question 3

Use a calculator to approximate the square root to 3 decimal places. Check to see that the approximation is reasonable.
-$\sqrt { 23 }$

Accepted Answer

A)  $4.796$ 
B)  $4.793$ 
C)  $23.000$ 
D)  $4.801$ 
A)  $4.796$ 
B)  $4.793$ 
C)  $23.000$ 
D)  $4.801$

Question 4

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\frac { 4 } { \sqrt { 6 } - 9 }$

Accepted Answer

A)  $\frac { 4 \sqrt { 6 } - 36 } { 75 }$ 
B)  $- \frac { 4 \sqrt { 6 } - 36 } { 75 }$ 
C)  $- \frac { 4 \sqrt { 6 } + 36 } { 75 }$ 
D)  $\frac { 4 \sqrt { 6 } + 36 } { 75 }$ 
A)  $\frac { 4 \sqrt { 6 } - 36 } { 75 }$ 
B)  $- \frac { 4 \sqrt { 6 } - 36 } { 75 }$ 
C)  $- \frac { 4 \sqrt { 6 } + 36 } { 75 }$ 
D)  $\frac { 4 \sqrt { 6 } + 36 } { 75 }$

Question 5

Write in terms of i.
-$- \sqrt { - 228 }$

Accepted Answer

A)  $2 \sqrt { 57 }$ 
B)  $- 2 \sqrt { 57 }$ 
C)  $- 2 \mathrm { i } \sqrt { 57 }$ 
D)  $2 \mathrm { i } \sqrt { 57 }$ 
A)  $2 \sqrt { 57 }$ 
B)  $- 2 \sqrt { 57 }$ 
C)  $- 2 \mathrm { i } \sqrt { 57 }$ 
D)  $2 \mathrm { i } \sqrt { 57 }$

Question 6

Rationalize the numerator and simplify. Assume all variables represent positive real numbers.
-$\frac { \sqrt { x } + 2 \sqrt { y } } { 3 \sqrt { x } }$

Accepted Answer

A)  $\frac { x + 4 y } { 3 x + 6 \sqrt { x y } }$ 
B)  $\frac { x - 2 y } { 3 x - 6 \sqrt { x y } }$ 
C)  $\frac { x - 4 y } { 3 x - 6 \sqrt { x y } }$ 
D)  $\frac { x - 4 y } { 3 x + 6 \sqrt { x y } }$ 
A)  $\frac { x + 4 y } { 3 x + 6 \sqrt { x y } }$ 
B)  $\frac { x - 2 y } { 3 x - 6 \sqrt { x y } }$ 
C)  $\frac { x - 4 y } { 3 x - 6 \sqrt { x y } }$ 
D)  $\frac { x - 4 y } { 3 x + 6 \sqrt { x y } }$

Question 7

Perform the indicated operation. Write the result in the form a + bi.
-$( 2 - 2 i ) + ( 8 + 8 i )$

Accepted Answer

A)  $- 6 + 10 \mathrm { i }$ 
B)  $10 - 6 i$ 
C)  $10 + 6 i$ 
D)  $- 10 - 6 i$ 
A)  $- 6 + 10 \mathrm { i }$ 
B)  $10 - 6 i$ 
C)  $10 + 6 i$ 
D)  $- 10 - 6 i$

Question 8

Find the cube root.
-$\sqrt [ 3 ] { - 8 x ^ { 30 } y ^ { 18 } }$

Accepted Answer

A)  $- 2 x ^ { 10 } y ^ { 6 }$ 
B)  $4 x ^ { 10 } y ^ { 6 }$ 
C)  $2 x ^ { 10 } y ^ { 6 }$ 
D)  $- 2 x ^ { 15 } y ^ { 9 }$ 
A)  $- 2 x ^ { 10 } y ^ { 6 }$ 
B)  $4 x ^ { 10 } y ^ { 6 }$ 
C)  $2 x ^ { 10 } y ^ { 6 }$ 
D)  $- 2 x ^ { 15 } y ^ { 9 }$

Question 9

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\sqrt { \frac { 11 x } { 2 y } }$

Accepted Answer

A)  $\sqrt { 22 x y }$ 
B)  $\frac { \sqrt { 22 x y } } { 2 }$ 
C)  $\frac { \sqrt { 22 x y } } { 2 y }$ 
D)  $\frac { \sqrt { 22 x y } } { 4 y ^ { 2 } }$ 
A)  $\sqrt { 22 x y }$ 
B)  $\frac { \sqrt { 22 x y } } { 2 }$ 
C)  $\frac { \sqrt { 22 x y } } { 2 y }$ 
D)  $\frac { \sqrt { 22 x y } } { 4 y ^ { 2 } }$

Question 10

Use radical notation to write the expression. Simplify if possible.
-$\left( - 8 x ^ { 9 } \right) ^ { 1 / 3 }$

Accepted Answer

A)  $- 8 x ^ { 3 }$ 
B)  $- 2 \sqrt [ 3 ] { x }$ 
C)  $- 2 x ^ { 9 }$ 
D)  $- 2 x ^ { 3 }$ 
A)  $- 8 x ^ { 3 }$ 
B)  $- 2 \sqrt [ 3 ] { x }$ 
C)  $- 2 x ^ { 9 }$ 
D)  $- 2 x ^ { 3 }$

Question 11

Solve.
-$\sqrt { x + 191 } - 5 = \sqrt { x + 76 }$

Accepted Answer

A)  $\varnothing$ 
B)  6 
C)  $- 5$ 
D)  5 
A)  $\varnothing$ 
B)  6 
C)  $- 5$ 
D)  5

Question 12

Factor the given factor from the expression.
-$x ^ { - 1 / 7 } ; x ^ { - 1 / 7 } + x ^ { 1 / 7 }$

Accepted Answer

A)  1 
B)  $x ^ { - 1 / 7 } \left( 1 + x ^ { 2 / 7 } \right)$ 
C)  $x ^ { 1 / 7 } \left( 1 + x ^ { 2 / 7 } \right)$ 
D)  0 
A)  1 
B)  $x ^ { - 1 / 7 } \left( 1 + x ^ { 2 / 7 } \right)$ 
C)  $x ^ { 1 / 7 } \left( 1 + x ^ { 2 / 7 } \right)$ 
D)  0

Question 13

Find the power of i.
-$( - 3 i ) ^ { 7 }$

Accepted Answer

A)  243 
B)  $- 2187 \mathrm { i }$ 
C)  $2187 \mathrm { i }$ 
D)  $- 243$ 
A)  243 
B)  $- 2187 \mathrm { i }$ 
C)  $2187 \mathrm { i }$ 
D)  $- 243$

Question 14

Simplify the radical expression. Assume that all variables represent positive real numbers.
-$\frac { \sqrt { 56 x ^ { 5 } y ^ { 6 } } } { \sqrt { 2 y ^ { 4 } } }$

Accepted Answer

A)  $4 x ^ { 2 } y \sqrt { 7 x }$ 
B)  $2 x ^ { 4 } y ^ { 2 } \sqrt { 7 x y }$ 
C)  $2 x ^ { 2 } y \sqrt { 7 x }$ 
D)  $28 x y \sqrt { x }$ 
A)  $4 x ^ { 2 } y \sqrt { 7 x }$ 
B)  $2 x ^ { 4 } y ^ { 2 } \sqrt { 7 x y }$ 
C)  $2 x ^ { 2 } y \sqrt { 7 x }$ 
D)  $28 x y \sqrt { x }$

Question 15

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

Accepted Answer

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

Question 16

Fill in the blank with one of the words or phrases listed below. $$\begin{array} { l l l } 
\text { index rationalizing } & \text { conjugate } & \text { principal square rootcube root midpoint } \
\text { complex numberlike radicals } & \text { radicand } & \text { imaginary unit } \quad \text { distance }
\end{array}$$
-Radicals with the same index and the same radicand are called --------

Accepted Answer

A)  rationalizing 
B)  distance 
C)  like radicals 
D)  midpoint 
A)  rationalizing 
B)  distance 
C)  like radicals 
D)  midpoint

Question 17

Evaluate.
-If $f ( x ) = \sqrt { 2 x - 9 }$, find the value of $f ( 29 )$.

Accepted Answer

A)  $\sqrt { 58 }$ 
B)  58 
C)  7 
D)  49 
A)  $\sqrt { 58 }$ 
B)  58 
C)  7 
D)  49

Question 18

Perform the indicated operation. Write the result in the form a + bi.
-(-9 + 2i) - 6

Accepted Answer

A)  -3 + 2i 
B)  15 - 2i 
C)  -15 + 2i 
D)  -3 - 2i 
A)  -3 + 2i 
B)  15 - 2i 
C)  -15 + 2i 
D)  -3 - 2i

Question 19

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\sqrt [ 3 ] { \frac { 5 } { 9 } }$

Accepted Answer

A)  $\frac { \sqrt [ 3 ] { 405 } } { 9 }$ 
B)  $\frac { \sqrt [ 3 ] { 405 } } { 81 }$ 
C)  $\frac { \sqrt [ 3 ] { 45 } } { 9 }$ 
D)  $\frac { \sqrt [ 3 ] { 15 } } { 3 }$ 
A)  $\frac { \sqrt [ 3 ] { 405 } } { 9 }$ 
B)  $\frac { \sqrt [ 3 ] { 405 } } { 81 }$ 
C)  $\frac { \sqrt [ 3 ] { 45 } } { 9 }$ 
D)  $\frac { \sqrt [ 3 ] { 15 } } { 3 }$

Question 20

Use the product rule to multiply. Assume all variables represent positive real numbers.
-$\sqrt { 10 x } \cdot \sqrt { 17 y }$

Accepted Answer

A)  $\sqrt { 10 x + 17 y }$ 
B)  $170 x y$ 
C)  $\sqrt { 27 x y }$ 
D)  $\sqrt { 170 x y }$ 
A)  $\sqrt { 10 x + 17 y }$ 
B)  $170 x y$ 
C)  $\sqrt { 27 x y }$ 
D)  $\sqrt { 170 x y }$

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - $( 6 \sqrt { 5 } + 3 ) ( 6 \sqrt { 5 } + 8 )$

Use rational exponents to write as a single radical expression. - $\sqrt [ 5 ] { 4 } \cdot \sqrt { 2 }$

Use a calculator to approximate the square root to 3 decimal places. Check to see that the approximation is reasonable. - $\sqrt { 23 }$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { 4 } { \sqrt { 6 } - 9 }$

Write in terms of i. - $- \sqrt { - 228 }$

Rationalize the numerator and simplify. Assume all variables represent positive real numbers. - $\frac { \sqrt { x } + 2 \sqrt { y } } { 3 \sqrt { x } }$

Perform the indicated operation. Write the result in the form a + bi. - $( 2 - 2 i ) + ( 8 + 8 i )$

Find the cube root. - $\sqrt [ 3 ] { - 8 x ^ { 30 } y ^ { 18 } }$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\sqrt { \frac { 11 x } { 2 y } }$

Use radical notation to write the expression. Simplify if possible. - $\left( - 8 x ^ { 9 } \right) ^ { 1 / 3 }$

Solve. - $\sqrt { x + 191 } - 5 = \sqrt { x + 76 }$

Factor the given factor from the expression. - $x ^ { - 1 / 7 } ; x ^ { - 1 / 7 } + x ^ { 1 / 7 }$

Find the power of i. - $( - 3 i ) ^ { 7 }$

Simplify the radical expression. Assume that all variables represent positive real numbers. - $\frac { \sqrt { 56 x ^ { 5 } y ^ { 6 } } } { \sqrt { 2 y ^ { 4 } } }$

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

Fill in the blank with one of the words or phrases listed below. index rationalizing conjugate principal square rootcube root midpoint complex numberlike radicals radicand imaginary unit distance -Radicals with the same index and the same radicand are called --------

Evaluate. -If $f ( x ) = \sqrt { 2 x - 9 }$ , find the value of $f ( 29 )$ .

Perform the indicated operation. Write the result in the form a + bi. -(-9 + 2i) - 6

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\sqrt [ 3 ] { \frac { 5 } { 9 } }$

Use the product rule to multiply. Assume all variables represent positive real numbers. - $\sqrt { 10 x } \cdot \sqrt { 17 y }$

Review of Real Numbers

Equations, Inequalities, and Problem Solving

Graphing

Solving Systems of Linear Equations

Exponents and Polynomials

Factoring Polynomials

Rational Expressions

More on Functions and Graphs

Inequalities and Absolute Value

Quadratic Equations and Functions

Exponential and Logarithmic Functions

Conic Sections

Sequences, Series, and the Binomial Theorem

Filters

Exam 10: Rational Exponents, Radicals, and Complex Numbers

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - (65+3)(65+8)( 6 \sqrt { 5 } + 3 ) ( 6 \sqrt { 5 } + 8 )(65​+3)(65​+8)

Use rational exponents to write as a single radical expression. - 45⋅2\sqrt [ 5 ] { 4 } \cdot \sqrt { 2 }54​⋅2​

Use a calculator to approximate the square root to 3 decimal places. Check to see that the approximation is reasonable. - 23\sqrt { 23 }23​

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - 46−9\frac { 4 } { \sqrt { 6 } - 9 }6​−94​

Write in terms of i. - −−228- \sqrt { - 228 }−−228​

Rationalize the numerator and simplify. Assume all variables represent positive real numbers. - x+2y3x\frac { \sqrt { x } + 2 \sqrt { y } } { 3 \sqrt { x } }3x​x​+2y​​

Perform the indicated operation. Write the result in the form a + bi. - (2−2i)+(8+8i)( 2 - 2 i ) + ( 8 + 8 i )(2−2i)+(8+8i)

Find the cube root. - −8x30y183\sqrt [ 3 ] { - 8 x ^ { 30 } y ^ { 18 } }3−8x30y18​

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - 11x2y\sqrt { \frac { 11 x } { 2 y } }2y11x​​

Use radical notation to write the expression. Simplify if possible. - (−8x9)1/3\left( - 8 x ^ { 9 } \right) ^ { 1 / 3 }(−8x9)1/3

Solve. - x+191−5=x+76\sqrt { x + 191 } - 5 = \sqrt { x + 76 }x+191​−5=x+76​

Factor the given factor from the expression. - x−1/7;x−1/7+x1/7x ^ { - 1 / 7 } ; x ^ { - 1 / 7 } + x ^ { 1 / 7 }x−1/7;x−1/7+x1/7

Find the power of i. - (−3i)7( - 3 i ) ^ { 7 }(−3i)7

Simplify the radical expression. Assume that all variables represent positive real numbers. - 56x5y62y4\frac { \sqrt { 56 x ^ { 5 } y ^ { 6 } } } { \sqrt { 2 y ^ { 4 } } }2y4​56x5y6​​

Rationalize the denominator. Assume that all variables represent positive numbers. - 4x\sqrt { \frac { 4 } { x } }x4​​