Question 1

Perform the indicated operation. Write the result in the form a + bi.
-7i + (-6 - i)

Accepted Answer

A)  -6 + 6i 
B)  -6 + 8i 
C)  6 - 8i 
D)  6 - 6i 
A)  -6 + 6i 
B)  -6 + 8i 
C)  6 - 8i 
D)  6 - 6i

Question 2

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

Accepted Answer

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

Question 3

Perform the indicated operation. Write the result in the form a + bi.
-$( 2 - 9 \mathrm { i } ) ^ { 2 }$

Accepted Answer

A)  $- 77 + 0 \mathrm { i }$ 
B)  $- 77 - 36 i$ 
C)  $85 - 36 i$ 
D)  $4 - 36 i + 81 i ^ { 2 }$ 
A)  $- 77 + 0 \mathrm { i }$ 
B)  $- 77 - 36 i$ 
C)  $85 - 36 i$ 
D)  $4 - 36 i + 81 i ^ { 2 }$

Question 4

Perform the indicated operations. Assume that all variables represent positive numbers.
-$( \sqrt { 13 } + 2 ) ( \sqrt { 13 } - 2 )$

Accepted Answer

A)  17 
B)  $13 - 2 \sqrt { 2 }$ 
C)  11 
D)  9 
A)  17 
B)  $13 - 2 \sqrt { 2 }$ 
C)  11 
D)  9

Question 5

Fill in the blank with one of the words or phrases listed below. $$\begin{array} { l l l } 
\text { index rationalizing } & \text { conjugate } & \text { principal square rootcube root midpoint } \
\text { complex numberlike radicals } & \text { radicand } & \text { imaginary unit } \quad \text { distance }
\end{array}$$
-The_____ formula is $\left( \frac { x _ { 1 } + x _ { 2 } } { 2 } , \frac { y _ { 1 } + y _ { 2 } } { 2 } \right)$.

Accepted Answer

A)  conjugate 
B)  midpoint 
C)  cube root 
D)  distance 
A)  conjugate 
B)  midpoint 
C)  cube root 
D)  distance

Question 6

Use a calculator to approximate the number to three decimal places.
-$534 ^ { - 2 / 3 }$

Accepted Answer

A)  $0.017$ 
B)  $0.012$ 
C)  $0.014$ 
D)  $0.015$ 
A)  $0.017$ 
B)  $0.012$ 
C)  $0.014$ 
D)  $0.015$

Question 7

Perform the indicated operation. Write the result in the form a + bi.
-$\frac { 6 } { 3 + i }$

Accepted Answer

A)  $\frac { 9 } { 4 } - \frac { 3 } { 4 } \mathrm { i }$ 
B)  $\frac { 9 } { 5 } + \frac { 3 } { 5 } \mathrm { i }$ 
C)  $\frac { 9 } { 5 } - \frac { 3 } { 5 } \mathrm { i }$ 
D)  $\frac { 9 } { 4 } + \frac { 3 } { 4 } \mathrm { i }$ 
A)  $\frac { 9 } { 4 } - \frac { 3 } { 4 } \mathrm { i }$ 
B)  $\frac { 9 } { 5 } + \frac { 3 } { 5 } \mathrm { i }$ 
C)  $\frac { 9 } { 5 } - \frac { 3 } { 5 } \mathrm { i }$ 
D)  $\frac { 9 } { 4 } + \frac { 3 } { 4 } \mathrm { i }$

Question 8

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

Accepted Answer

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

Question 9

Use radical notation to write the expression. Simplify if possible.
-$243 ^ { 4 / 5 }$

Accepted Answer

A)  2187 
B)  19,683 
C)  6561 
D)  81 
A)  2187 
B)  19,683 
C)  6561 
D)  81

Question 10

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

Accepted Answer

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

Question 11

Simplify the radical expression. Assume that all variables represent positive real numbers.
-$\sqrt { 98 x ^ { 2 } }$

Accepted Answer

A)  $98 x$ 
B)  $7x \sqrt { 2 }$ 
C)  $7 \sqrt { 2 x }$ 
D)  $2 x ^ { 2 } \sqrt { 7 }$ 
A)  $98 x$ 
B)  $7x \sqrt { 2 }$ 
C)  $7 \sqrt { 2 x }$ 
D)  $2 x ^ { 2 } \sqrt { 7 }$

Question 12

Use rational exponents to write as a single radical expression.
-$\frac { \sqrt [ 7 ] { \mathrm { y } } } { \sqrt [ 8 ] { \mathrm { y } } }$

Accepted Answer

A)  $\sqrt [ 8 ] { \mathrm { y } ^ { 7 } }$ 
B)  $\sqrt [ 56 ] { \mathrm { y } }$ 
C)  $y$ 
D)  $\frac { 1 } { \sqrt { y } }$ 
A)  $\sqrt [ 8 ] { \mathrm { y } ^ { 7 } }$ 
B)  $\sqrt [ 56 ] { \mathrm { y } }$ 
C)  $y$ 
D)  $\frac { 1 } { \sqrt { y } }$

Question 13

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

Accepted Answer

A)  $\frac { 15 + 3 \sqrt { x } } { x - 25 }$ 
B)  $\frac { 15 + 3 \sqrt { x } } { x + 25 }$ 
C)  $\frac { 15 - 3 \sqrt { x } } { x ^ { 2 } - 25 }$ 
D)  $\frac { 15 - 3 \sqrt { x } } { x - 25 }$ 
A)  $\frac { 15 + 3 \sqrt { x } } { x - 25 }$ 
B)  $\frac { 15 + 3 \sqrt { x } } { x + 25 }$ 
C)  $\frac { 15 - 3 \sqrt { x } } { x ^ { 2 } - 25 }$ 
D)  $\frac { 15 - 3 \sqrt { x } } { x - 25 }$

Question 14

Add or subtract. Assume all variables represent positive real numbers.
-$\sqrt { 9 } + \sqrt { 180 } + \sqrt { 4 } + \sqrt { 245 }$

Accepted Answer

A)  $85 \sqrt { 5 } + 5$ 
B)  $\sqrt { 180 } + \sqrt { 245 } + 5$ 
C)  $13 \sqrt { 5 } + \sqrt { 9 } + \sqrt { 4 }$ 
D)  $13 \sqrt { 5 } + 5$ 
A)  $85 \sqrt { 5 } + 5$ 
B)  $\sqrt { 180 } + \sqrt { 245 } + 5$ 
C)  $13 \sqrt { 5 } + \sqrt { 9 } + \sqrt { 4 }$ 
D)  $13 \sqrt { 5 } + 5$

Question 15

Find the root. Assume that all variables represent nonnegative real numbers.
-$\sqrt [ 5 ] { \frac { 1024 x ^ { 5 } } { y ^ { 20 } } }$

Accepted Answer

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

Question 16

Solve.
-Scott set up a volleyball net in his backyard. One of the poles, which forms a right angle with the ground, is 7 feet high. To secure the pole, he attached a rope from the top of the pole to a stake 5 feet from the bottom of the pole. To the nearest tenth of a foot, find the length of the rope.

Accepted Answer

A)  $74 \mathrm { ft }$ 
B)  $4.9 \mathrm { ft }$ 
C)  $3.5 \mathrm { ft }$ 
D)  $8.6 \mathrm { ft }$ 
A)  $74 \mathrm { ft }$ 
B)  $4.9 \mathrm { ft }$ 
C)  $3.5 \mathrm { ft }$ 
D)  $8.6 \mathrm { ft }$

Question 17

Use the product rule to multiply. Assume all variables represent positive real numbers.
-$\sqrt [ 3 ] { 343 } \cdot \sqrt [ 3 ] { 729 }$

Accepted Answer

A)  $- 2$ 
B)  63 
C)  16 
D)  $- 63$ 
A)  $- 2$ 
B)  63 
C)  16 
D)  $- 63$

Question 18

Find the midpoint of the line segment whose endpoints are given.
-$( 7 \sqrt { 6 } , 8 \sqrt { 2 } ) , ( 10 \sqrt { 6 } , 11 \sqrt { 2 } )$

Accepted Answer

A)  $\left( \frac { 3 \sqrt { 6 } } { 2 } , \frac { 3 \sqrt { 2 } } { 2 } \right)$ 
B)  $\left( \frac { 17 \sqrt { 6 } } { 2 } , \frac { 19 \sqrt { 2 } } { 2 } \right)$ 
C)  $\left( \frac { - 3 \sqrt { 6 } } { 2 } , \frac { - 3 \sqrt { 2 } } { 2 } \right)$ 
D)  $( 17 \sqrt { 6 } , 19 \sqrt { 2 } )$ 
A)  $\left( \frac { 3 \sqrt { 6 } } { 2 } , \frac { 3 \sqrt { 2 } } { 2 } \right)$ 
B)  $\left( \frac { 17 \sqrt { 6 } } { 2 } , \frac { 19 \sqrt { 2 } } { 2 } \right)$ 
C)  $\left( \frac { - 3 \sqrt { 6 } } { 2 } , \frac { - 3 \sqrt { 2 } } { 2 } \right)$ 
D)  $( 17 \sqrt { 6 } , 19 \sqrt { 2 } )$

Question 19

Perform the indicated operation. Write the result in the form a + bi.
-$\frac { 7 + 2 i } { 7 - 2 i }$

Accepted Answer

A)  $\frac { 53 } { 45 } - \frac { 28 } { 45 } \mathrm { i }$ 
B)  $1 + \frac { 28 } { 45 } \mathrm { i }$ 
C)  $1 + \frac { 28 } { 53 } \mathrm { i }$ 
D)  $\frac { 45 } { 53 } + \frac { 28 } { 53 } \mathrm { i }$ 
A)  $\frac { 53 } { 45 } - \frac { 28 } { 45 } \mathrm { i }$ 
B)  $1 + \frac { 28 } { 45 } \mathrm { i }$ 
C)  $1 + \frac { 28 } { 53 } \mathrm { i }$ 
D)  $\frac { 45 } { 53 } + \frac { 28 } { 53 } \mathrm { i }$

Question 20

Find the midpoint of the line segment whose endpoints are given.
-$\left( - \frac { 5 } { 2 } , 1 \right) , \left( 4 , - \frac { 9 } { 5 } \right)$

Accepted Answer

A)  $\left( \frac { 3 } { 2 } , - \frac { 4 } { 5 } \right)$ 
B)  $\left( \frac { 3 } { 4 } , - \frac { 2 } { 5 } \right)$ 
C)  $\left( - \frac { 13 } { 4 } , \frac { 7 } { 5 } \right)$ 
D)  $\left( \frac { 13 } { 4 } , - \frac { 7 } { 5 } \right)$ 
A)  $\left( \frac { 3 } { 2 } , - \frac { 4 } { 5 } \right)$ 
B)  $\left( \frac { 3 } { 4 } , - \frac { 2 } { 5 } \right)$ 
C)  $\left( - \frac { 13 } { 4 } , \frac { 7 } { 5 } \right)$ 
D)  $\left( \frac { 13 } { 4 } , - \frac { 7 } { 5 } \right)$

Perform the indicated operation. Write the result in the form a + bi. -7i + (-6 - i)

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

Perform the indicated operation. Write the result in the form a + bi. - $( 2 - 9 \mathrm { i } ) ^ { 2 }$

Perform the indicated operations. Assume that all variables represent positive numbers. - $( \sqrt { 13 } + 2 ) ( \sqrt { 13 } - 2 )$

Use a calculator to approximate the number to three decimal places. - $534 ^ { - 2 / 3 }$

Perform the indicated operation. Write the result in the form a + bi. - $\frac { 6 } { 3 + i }$

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

Use radical notation to write the expression. Simplify if possible. - $243 ^ { 4 / 5 }$

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

Simplify the radical expression. Assume that all variables represent positive real numbers. - $\sqrt { 98 x ^ { 2 } }$

Use rational exponents to write as a single radical expression. - $\frac { \sqrt [ 7 ] { \mathrm { y } } } { \sqrt [ 8 ] { \mathrm { y } } }$

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

Add or subtract. Assume all variables represent positive real numbers. - $\sqrt { 9 } + \sqrt { 180 } + \sqrt { 4 } + \sqrt { 245 }$

Find the root. Assume that all variables represent nonnegative real numbers. - $\sqrt [ 5 ] { \frac { 1024 x ^ { 5 } } { y ^ { 20 } } }$

Use the product rule to multiply. Assume all variables represent positive real numbers. - $\sqrt [ 3 ] { 343 } \cdot \sqrt [ 3 ] { 729 }$

Find the midpoint of the line segment whose endpoints are given. - $( 7 \sqrt { 6 } , 8 \sqrt { 2 } ) , ( 10 \sqrt { 6 } , 11 \sqrt { 2 } )$

Perform the indicated operation. Write the result in the form a + bi. - $\frac { 7 + 2 i } { 7 - 2 i }$

Find the midpoint of the line segment whose endpoints are given. - $\left( - \frac { 5 } { 2 } , 1 \right) , \left( 4 , - \frac { 9 } { 5 } \right)$

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

Perform the indicated operation. Write the result in the form a + bi. -7i + (-6 - i)

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

Perform the indicated operation. Write the result in the form a + bi. - (2−9i)2( 2 - 9 \mathrm { i } ) ^ { 2 }(2−9i)2

Perform the indicated operations. Assume that all variables represent positive numbers. - (13+2)(13−2)( \sqrt { 13 } + 2 ) ( \sqrt { 13 } - 2 )(13​+2)(13​−2)

Use a calculator to approximate the number to three decimal places. - 534−2/3534 ^ { - 2 / 3 }534−2/3

Perform the indicated operation. Write the result in the form a + bi. - 63+i\frac { 6 } { 3 + i }3+i6​

Write with positive exponents. Simplify if possible. - 243−4/5243 ^ { - 4 / 5 }243−4/5

Use radical notation to write the expression. Simplify if possible. - 2434/5243 ^ { 4 / 5 }2434/5

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

Simplify the radical expression. Assume that all variables represent positive real numbers. - 98x2\sqrt { 98 x ^ { 2 } }98x2​

Use rational exponents to write as a single radical expression. - y7y8\frac { \sqrt [ 7 ] { \mathrm { y } } } { \sqrt [ 8 ] { \mathrm { y } } }8y​7y​​

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - −3x+5\frac { - 3 } { \sqrt { x } + 5 }x​+5−3​

Add or subtract. Assume all variables represent positive real numbers. - 9+180+4+245\sqrt { 9 } + \sqrt { 180 } + \sqrt { 4 } + \sqrt { 245 }9​+180​+4​+245​

Find the root. Assume that all variables represent nonnegative real numbers. - 1024x5y205\sqrt [ 5 ] { \frac { 1024 x ^ { 5 } } { y ^ { 20 } } }5y201024x5​​

Use the product rule to multiply. Assume all variables represent positive real numbers. - 3433⋅7293\sqrt [ 3 ] { 343 } \cdot \sqrt [ 3 ] { 729 }3343​⋅3729​

Find the midpoint of the line segment whose endpoints are given. - (76,82),(106,112)( 7 \sqrt { 6 } , 8 \sqrt { 2 } ) , ( 10 \sqrt { 6 } , 11 \sqrt { 2 } )(76​,82​),(106​,112​)

Perform the indicated operation. Write the result in the form a + bi. - 7+2i7−2i\frac { 7 + 2 i } { 7 - 2 i }7−2i7+2i​

Find the midpoint of the line segment whose endpoints are given. - (−52,1),(4,−95)\left( - \frac { 5 } { 2 } , 1 \right) , \left( 4 , - \frac { 9 } { 5 } \right)(−25​,1),(4,−59​)