Question 1

Perform the indicated operation and simplify. Write the result in the form a $a + bi$
-$( 2 + 4 \mathrm { i } ) ( 2 - 4 \mathrm { i } )$

Accepted Answer

A)  $20 + 0 \mathrm { i }$ 
B)  $4 - 16 i$ 
C)  $- 12 + 0 \mathrm { i }$ 
D)  $4 - 16 \mathrm { i } ^ { 2 }$ 
A)  $20 + 0 \mathrm { i }$ 
B)  $4 - 16 i$ 
C)  $- 12 + 0 \mathrm { i }$ 
D)  $4 - 16 \mathrm { i } ^ { 2 }$

Question 2

Simplify the radical expression. Assume that all variables represent positive real numbers.
-$\sqrt [ 3 ] { - 27 a ^ { 11 } b ^ { 13 } }$

Accepted Answer

A)  $- 3 a ^ { 3 } b ^ { 4 } \sqrt [ 3 ] { a ^ { 2 } b }$ 
B)  $3 a b \sqrt [ 3 ] { a ^ { 5 } b ^ { 4 } }$ 
C)  $3 a ^ { 2 } b \sqrt [ 3 ] { a ^ { 3 } b ^ { 4 } }$ 
D)  $3 \sqrt { \mathrm { a } ^ { 13 } \mathrm {~b} ^ { 11 } }$ 
A)  $- 3 a ^ { 3 } b ^ { 4 } \sqrt [ 3 ] { a ^ { 2 } b }$ 
B)  $3 a b \sqrt [ 3 ] { a ^ { 5 } b ^ { 4 } }$ 
C)  $3 a ^ { 2 } b \sqrt [ 3 ] { a ^ { 3 } b ^ { 4 } }$ 
D)  $3 \sqrt { \mathrm { a } ^ { 13 } \mathrm {~b} ^ { 11 } }$

Question 3

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

Accepted Answer

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

Question 4

Solve.
-$\sqrt { 3 x + 1 } = 3 + \sqrt { x - 4 }$

Accepted Answer

A)  $\varnothing$ 
B)  $- 8 , - 5$ 
C)  5,8 
D)  $- 5,8$ 
A)  $\varnothing$ 
B)  $- 8 , - 5$ 
C)  5,8 
D)  $- 5,8$

Question 5

Use the product rule to multiply. Assume all variables represent positive real numbers.
-$\sqrt { 28 } \cdot \sqrt { 7 }$

Accepted Answer

A)  196 
B)  14 
C)  $2 \sqrt { 7 }$ 
D)  $7 \sqrt { 2 }$ 
A)  196 
B)  14 
C)  $2 \sqrt { 7 }$ 
D)  $7 \sqrt { 2 }$

Question 6

Solve.
-$\sqrt [ 3 ] { 5 x + 3 } - 3 = 0$

Accepted Answer

A)  $\frac { 27 } { 5 }$ 
B)  $\frac { 6 } { 5 }$ 
C)  $\frac { 22 } { 5 }$ 
D)  $\frac { 24 } { 5 }$ 
A)  $\frac { 27 } { 5 }$ 
B)  $\frac { 6 } { 5 }$ 
C)  $\frac { 22 } { 5 }$ 
D)  $\frac { 24 } { 5 }$

Question 7

Multiply, and then simplify if possible. Assume all variables represent positive real numbers.
-$( \sqrt { 77 } - \sqrt { 847 } ) ( \sqrt { 11 } + \sqrt { 7 } )$

Accepted Answer

A)  $11 \sqrt { 7 } + 7 \sqrt { 11 } - 11 \sqrt { 77 } + 77$ 
B)  $11 \sqrt { 7 } + 7 \sqrt { 11 } - 11 \sqrt { 77 } - 77$ 
C)  $11 \sqrt { 7 } + 7 \sqrt { 11 } + 11 \sqrt { 77 } + 77$ 
D)  $11 \sqrt { 7 } - 7 \sqrt { 11 } + 11 \sqrt { 77 } + 77$ 
A)  $11 \sqrt { 7 } + 7 \sqrt { 11 } - 11 \sqrt { 77 } + 77$ 
B)  $11 \sqrt { 7 } + 7 \sqrt { 11 } - 11 \sqrt { 77 } - 77$ 
C)  $11 \sqrt { 7 } + 7 \sqrt { 11 } + 11 \sqrt { 77 } + 77$ 
D)  $11 \sqrt { 7 } - 7 \sqrt { 11 } + 11 \sqrt { 77 } + 77$

Question 8

Solve.
-Find the area of the rectangle.

Accepted Answer

A)  $60 \mathrm { sq }$. $\mathrm { ft }$ 
B)  $30 \mathrm { sq } . \mathrm { ft }$ 
C)  $15 \sqrt { 2 } \mathrm { sq } . \mathrm { ft }$ 
D)  $30 \sqrt { 2 } \mathrm { sq }$. $\mathrm { ft }$ 
A)  $60 \mathrm { sq }$. $\mathrm { ft }$ 
B)  $30 \mathrm { sq } . \mathrm { ft }$ 
C)  $15 \sqrt { 2 } \mathrm { sq } . \mathrm { ft }$ 
D)  $30 \sqrt { 2 } \mathrm { sq }$. $\mathrm { ft }$

Question 9

Simplify. Assume that all variables represent any real number.
-$\sqrt [ 5 ] { \mathrm { x } ^ { 5 } }$

Accepted Answer

A)  $- | x |$ 
B)  $| x |$ 
C)  $x$ 
D)  $- x$ 
A)  $- | x |$ 
B)  $| x |$ 
C)  $x$ 
D)  $- x$

Question 10

Find the root. Assume that all variables represent nonnegative real numbers.
-$\sqrt [ 4 ] { 16 x ^ { 8 } y ^ { 16 } }$

Accepted Answer

A)  $2 x ^ { 2 } y ^ { 4 }$ 
B)  not a real number 
C)  $2.497 x ^ { 2 } y ^ { 4 }$ 
D)  $4 \mathrm { x } ^ { 4 } \mathrm { y } ^ { 2 }$ 
A)  $2 x ^ { 2 } y ^ { 4 }$ 
B)  not a real number 
C)  $2.497 x ^ { 2 } y ^ { 4 }$ 
D)  $4 \mathrm { x } ^ { 4 } \mathrm { y } ^ { 2 }$

Question 11

Use the properties of exponents to simplify the expression. Write with positive exponents.
-$\left( r ^ { 1 / 3 } \cdot s ^ { 1 / 3 } \right) ^ { 2 }$

Accepted Answer

A)  $r ^ { 1 / 6 } s ^ { 1 / 6 }$ 
B)  $r ^ { 1 / 9 } s ^ { 1 / 9 }$ 
C)  $r ^ { 2 } s ^ { 2 }$ 
D)  $\mathrm { r } ^ { 2 / 3 } \mathrm {~s} ^ { 2 / 3 }$ 
A)  $r ^ { 1 / 6 } s ^ { 1 / 6 }$ 
B)  $r ^ { 1 / 9 } s ^ { 1 / 9 }$ 
C)  $r ^ { 2 } s ^ { 2 }$ 
D)  $\mathrm { r } ^ { 2 / 3 } \mathrm {~s} ^ { 2 / 3 }$

Question 12

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

Accepted Answer

A)  $9 \mathrm { y }$ 
B)  $\frac { 9 \sqrt [ 3 ] { y } } { y }$ 
C)  $\frac { 9 \sqrt [ 3 ] { y ^ { 2 } } } { y }$ 
D)  $9 \sqrt [ 3 ] { \mathrm { y } ^ { 2 } }$ 
A)  $9 \mathrm { y }$ 
B)  $\frac { 9 \sqrt [ 3 ] { y } } { y }$ 
C)  $\frac { 9 \sqrt [ 3 ] { y ^ { 2 } } } { y }$ 
D)  $9 \sqrt [ 3 ] { \mathrm { y } ^ { 2 } }$

Question 13

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

Accepted Answer

A) $3 \sqrt{2}$ 
B)  $2 \sqrt { 3 }$ 
C)  $\sqrt { 14 }$ 
D)  14 
A) $3 \sqrt{2}$ 
B)  $2 \sqrt { 3 }$ 
C)  $\sqrt { 14 }$ 
D)  14

Question 14

Find the cube root.
-$\sqrt [ 3 ] { - 64 x ^ { 6 } }$

Accepted Answer

A)  $16 x ^ { 2 }$ 
B)  $- 4 x ^ { 2 }$ 
C)  not a real number 
D)  $8 x ^ { 2 }$ 
A)  $16 x ^ { 2 }$ 
B)  $- 4 x ^ { 2 }$ 
C)  not a real number 
D)  $8 x ^ { 2 }$

Question 15

Find the cube root.
-$\sqrt [ 3 ] { \frac { 1 } { 216 } }$

Accepted Answer

A)  6 
B)  $\frac { 1 } { 36 }$ 
C)  $\frac { 1 } { 6 }$ 
D)  $\frac { 1 } { \sqrt [ 3 ] { 6 } }$ 
A)  6 
B)  $\frac { 1 } { 36 }$ 
C)  $\frac { 1 } { 6 }$ 
D)  $\frac { 1 } { \sqrt [ 3 ] { 6 } }$

Question 16

Solve.
-The maximum number of volts, $E$, that can be placed across a resistor is given by the formula $E = \sqrt { P R }$, where $P$ is the number of watts of power that the resistor can absorb and $R$ is the resistance of the resistor in ohms. If a 2-watt resistor can have at most 20 volts of electricity across it, find the number of ohms of resistance of this resistor.

Accepted Answer

A)  $800 \mathrm { ohms }$ 
B)  $100 \mathrm { ohms }$ 
C)  $200 \mathrm { ohms }$ 
D)  $10 \mathrm { ohms }$ 
A)  $800 \mathrm { ohms }$ 
B)  $100 \mathrm { ohms }$ 
C)  $200 \mathrm { ohms }$ 
D)  $10 \mathrm { ohms }$

Question 17

Multiply or divide.
-$\sqrt { - 9 } \cdot \sqrt { - 8 }$

Accepted Answer

A)  $- 6 \mathrm { i } \sqrt { 2 }$ 
B)  $- 6 \sqrt { 2 }$ 
C)  $6 \mathrm { i } \sqrt { 2 }$ 
D)  $6 \sqrt { 2 }$ 
A)  $- 6 \mathrm { i } \sqrt { 2 }$ 
B)  $- 6 \sqrt { 2 }$ 
C)  $6 \mathrm { i } \sqrt { 2 }$ 
D)  $6 \sqrt { 2 }$

Question 18

Use the product rule to multiply. Assume all variables represent positive real numbers.
-$\sqrt [ 5 ] { 5 x ^ { 4 } } \cdot \sqrt [ 5 ] { 81 }$

Accepted Answer

A)  $405 x ^ { 4 }$ 
B)  $9 x ^ { 2 }$ 
C)  $\sqrt { 405 x ^ { 4 } }$ 
D)  $\sqrt [ 5 ] { 405 x ^ { 4 } }$ 
A)  $405 x ^ { 4 }$ 
B)  $9 x ^ { 2 }$ 
C)  $\sqrt { 405 x ^ { 4 } }$ 
D)  $\sqrt [ 5 ] { 405 x ^ { 4 } }$

Question 19

Solve.
-The maximum number of volts, E, that can be placed across a resistor is given by the formula $\mathrm { E } = \sqrt { \mathrm { PR } }$, where $\mathrm { P }$ is the number of watts of power that the resistor can absorb and $R$ is the resistance of the resistor in ohms. If a $\frac { 1 } { 4 }$-watt resistor has a resistance of 10,000 ohms, find the largest number of volts of electricity that could be placed across the resistor.

Accepted Answer

A)  50 volts 
B)  25 volts 
C)  100 volts 
D)  2500 volts 
A)  50 volts 
B)  25 volts 
C)  100 volts 
D)  2500 volts

Question 20

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\sqrt [ 4 ] { \frac { 81 } { 121 x ^ { 9 } } }$

Accepted Answer

A)  $\frac { 3 \sqrt [ 4 ] { 121 x ^ { 3 } } } { 11 x ^ { 3 } }$ 
B)  $\frac { 3 \sqrt [ 4 ] { 121 x ^ { 3 } } } { 11 x ^ { 2 } }$ 
C)  $\frac { 3 \sqrt [ 4 ] { 121 x } } { 11 x ^ { 3 } }$ 
D)  $\frac { 3 \sqrt [ 4 ] { 121 x } } { 11 x ^ { 2 } }$ 
A)  $\frac { 3 \sqrt [ 4 ] { 121 x ^ { 3 } } } { 11 x ^ { 3 } }$ 
B)  $\frac { 3 \sqrt [ 4 ] { 121 x ^ { 3 } } } { 11 x ^ { 2 } }$ 
C)  $\frac { 3 \sqrt [ 4 ] { 121 x } } { 11 x ^ { 3 } }$ 
D)  $\frac { 3 \sqrt [ 4 ] { 121 x } } { 11 x ^ { 2 } }$

Perform the indicated operation and simplify. Write the result in the form a $a + bi$ - $( 2 + 4 \mathrm { i } ) ( 2 - 4 \mathrm { i } )$

Simplify the radical expression. Assume that all variables represent positive real numbers. - $\sqrt [ 3 ] { - 27 a ^ { 11 } b ^ { 13 } }$

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

Solve. - $\sqrt { 3 x + 1 } = 3 + \sqrt { x - 4 }$

Use the product rule to multiply. Assume all variables represent positive real numbers. - $\sqrt { 28 } \cdot \sqrt { 7 }$

Solve. - $\sqrt [ 3 ] { 5 x + 3 } - 3 = 0$

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - $( \sqrt { 77 } - \sqrt { 847 } ) ( \sqrt { 11 } + \sqrt { 7 } )$

Solve. -Find the area of the rectangle.

Simplify. Assume that all variables represent any real number. - $\sqrt [ 5 ] { \mathrm { x } ^ { 5 } }$

Find the root. Assume that all variables represent nonnegative real numbers. - $\sqrt [ 4 ] { 16 x ^ { 8 } y ^ { 16 } }$

Use the properties of exponents to simplify the expression. Write with positive exponents. - $\left( r ^ { 1 / 3 } \cdot s ^ { 1 / 3 } \right) ^ { 2 }$

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

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

Find the cube root. - $\sqrt [ 3 ] { - 64 x ^ { 6 } }$

Find the cube root. - $\sqrt [ 3 ] { \frac { 1 } { 216 } }$

Multiply or divide. - $\sqrt { - 9 } \cdot \sqrt { - 8 }$

Use the product rule to multiply. Assume all variables represent positive real numbers. - $\sqrt [ 5 ] { 5 x ^ { 4 } } \cdot \sqrt [ 5 ] { 81 }$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\sqrt [ 4 ] { \frac { 81 } { 121 x ^ { 9 } } }$

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 and simplify. Write the result in the form a a+bia + bia+bi - (2+4i)(2−4i)( 2 + 4 \mathrm { i } ) ( 2 - 4 \mathrm { i } )(2+4i)(2−4i)

Simplify the radical expression. Assume that all variables represent positive real numbers. - −27a11b133\sqrt [ 3 ] { - 27 a ^ { 11 } b ^ { 13 } }3−27a11b13​

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

Solve. - 3x+1=3+x−4\sqrt { 3 x + 1 } = 3 + \sqrt { x - 4 }3x+1​=3+x−4​

Use the product rule to multiply. Assume all variables represent positive real numbers. - 28⋅7\sqrt { 28 } \cdot \sqrt { 7 }28​⋅7​

Solve. - 5x+33−3=0\sqrt [ 3 ] { 5 x + 3 } - 3 = 035x+3​−3=0

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - (77−847)(11+7)( \sqrt { 77 } - \sqrt { 847 } ) ( \sqrt { 11 } + \sqrt { 7 } )(77​−847​)(11​+7​)