Question 1

Solve.
-$\sqrt { 6 x + 3 } = \sqrt { 4 x + 3 } - 1$

Accepted Answer

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

Question 2

Perform the indicated operation. Write the result in the form a + bi.
-$\frac { 7 } { 4 + 8 \mathrm { i } }$

Accepted Answer

A)  $\frac { 7 } { 20 } - \frac { 7 } { 10 } \mathrm { i }$ 
B)  $- \frac { 7 } { 12 } - \frac { 7 } { 6 } \mathrm { i }$ 
C)  $- \frac { 7 } { 12 } + \frac { 7 } { 6 } \mathrm { i }$ 
D)  $\frac { 7 } { 20 } + \frac { 7 } { 10 } \mathrm { i }$ 
A)  $\frac { 7 } { 20 } - \frac { 7 } { 10 } \mathrm { i }$ 
B)  $- \frac { 7 } { 12 } - \frac { 7 } { 6 } \mathrm { i }$ 
C)  $- \frac { 7 } { 12 } + \frac { 7 } { 6 } \mathrm { i }$ 
D)  $\frac { 7 } { 20 } + \frac { 7 } { 10 } \mathrm { i }$

Question 3

Use rational exponents to write as a single radical expression.
-$\sqrt [ 5 ] { 2 } \cdot \sqrt [ 3 ] { 3 }$

Accepted Answer

A)  $\sqrt [ 15 ] { 6 }$ 
B)  $\sqrt [ 5 ] { 6 }$ 
C)  $\sqrt [ 15 ] { 1944 }$ 
D)  $\sqrt [ 15 ] { 486 }$ 
A)  $\sqrt [ 15 ] { 6 }$ 
B)  $\sqrt [ 5 ] { 6 }$ 
C)  $\sqrt [ 15 ] { 1944 }$ 
D)  $\sqrt [ 15 ] { 486 }$

Question 4

Find the square root. Assume that all variables represent positive real numbers.
-$\sqrt { - 484 }$

Accepted Answer

A)  $- 22$ 
B)  $- 242$ 
C)  not a real number 
D)  22 
A)  $- 22$ 
B)  $- 242$ 
C)  not a real number 
D)  22

Question 5

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\frac { 6 \sqrt { 2 } + \sqrt { 6 } } { 3 \sqrt { 2 } - \sqrt { 6 } }$

Accepted Answer

A)  $\frac { 7 + 3 \sqrt { 3 } } { 2 }$ 
B)  $\frac { 42 + 9 \sqrt { 12 } } { 12 }$ 
C)  $\frac { 42 + 18 \sqrt { 3 } } { 12 }$ 
D)  $\frac { 21 + 9 \sqrt { 3 } } { 6 }$ 
A)  $\frac { 7 + 3 \sqrt { 3 } } { 2 }$ 
B)  $\frac { 42 + 9 \sqrt { 12 } } { 12 }$ 
C)  $\frac { 42 + 18 \sqrt { 3 } } { 12 }$ 
D)  $\frac { 21 + 9 \sqrt { 3 } } { 6 }$

Question 6

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

Accepted Answer

A)  16 
B)  $25 - \sqrt [ 3 ] { 9 }$ 
C)  22 
D)  $25 - \sqrt [ 3 ] { 3 }$ 
A)  16 
B)  $25 - \sqrt [ 3 ] { 9 }$ 
C)  22 
D)  $25 - \sqrt [ 3 ] { 3 }$

Question 7

Use the properties of exponents to simplify the expression. Write with positive exponents.
-$x ^ { 1 / 8 } \cdot x ^ { 7 / 8 }$

Accepted Answer

A)  $\frac { 1 } { x }$ 
B)  $x ^ { 7 / 64 }$ 
C)  $x ^ { 7 / 8 }$ 
D)  $x$ 
A)  $\frac { 1 } { x }$ 
B)  $x ^ { 7 / 64 }$ 
C)  $x ^ { 7 / 8 }$ 
D)  $x$

Question 8

Use the quotient rule to divide and simplify.
-$\sqrt { \frac { 32 x ^ { 2 } y } { 25 } }$

Accepted Answer

A)  $\frac { 4 \sqrt { 2 x ^ { 2 } y } } { 5 }$ 
B)  $16 x \sqrt { 2 y }$ 
C)  $x \sqrt { \frac { 32 y } { 5 } }$ 
D)  $\frac { 4 x \sqrt { 2 y } } { 5 }$ 
A)  $\frac { 4 \sqrt { 2 x ^ { 2 } y } } { 5 }$ 
B)  $16 x \sqrt { 2 y }$ 
C)  $x \sqrt { \frac { 32 y } { 5 } }$ 
D)  $\frac { 4 x \sqrt { 2 y } } { 5 }$

Question 9

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

Accepted Answer

A)  $7 \sqrt { 29 }$ 
B)  203 
C)  147 
D)  $7 \sqrt { 10 }$ 
A)  $7 \sqrt { 29 }$ 
B)  203 
C)  147 
D)  $7 \sqrt { 10 }$

Question 10

Find the distance between the pair of points.
-$( 2 , - 5 )$ and $( 6 , - 7 )$

Accepted Answer

A)  12 units 
B)  $12 \sqrt { 3 }$ units 
C)  6 units 
D)  $2 \sqrt { 5 }$ units 
A)  12 units 
B)  $12 \sqrt { 3 }$ units 
C)  6 units 
D)  $2 \sqrt { 5 }$ units

Question 11

Write with positive exponents. Simplify if possible.
-$16 ^ { - 5 / 4 }$

Accepted Answer

A)  $\frac { 1 } { 32 }$ 
B)  not a real number 
C)  $- \frac { 1 } { 32 }$ 
D)  32 
A)  $\frac { 1 } { 32 }$ 
B)  not a real number 
C)  $- \frac { 1 } { 32 }$ 
D)  32

Question 12

Solve.
-$\sqrt { x + 3 } = 8$

Accepted Answer

A)  64 
B)  67 
C)  121 
D)  61 
A)  64 
B)  67 
C)  121 
D)  61

Question 13

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\frac { 8 x } { \sqrt [ 5 ] { 27 x ^ { 17 } y ^ { 23 } } }$

Accepted Answer

A)  $\frac { 8 \sqrt [ 5 ] { 27 x ^ { 17 } y ^ { 23 } } } { 27 x ^ { 17 } y ^ { 23 } }$ 
B)  $\frac { 8 \sqrt [ 5 ] { 27 x ^ { 2 } y ^ { 3 } } } { 3 x ^ { 3 } y ^ { 4 } }$ 
C)  $\frac { 8 \sqrt [ 5 ] { 9 x ^ { 3 } y ^ { 2 } } } { 3 x ^ { 4 } y ^ { 5 } }$ 
D)  $\frac { 8 \sqrt [ 5 ] { 9 x ^ { 3 } y ^ { 2 } } } { 3 x ^ { 3 } y ^ { 4 } }$ 
A)  $\frac { 8 \sqrt [ 5 ] { 27 x ^ { 17 } y ^ { 23 } } } { 27 x ^ { 17 } y ^ { 23 } }$ 
B)  $\frac { 8 \sqrt [ 5 ] { 27 x ^ { 2 } y ^ { 3 } } } { 3 x ^ { 3 } y ^ { 4 } }$ 
C)  $\frac { 8 \sqrt [ 5 ] { 9 x ^ { 3 } y ^ { 2 } } } { 3 x ^ { 4 } y ^ { 5 } }$ 
D)  $\frac { 8 \sqrt [ 5 ] { 9 x ^ { 3 } y ^ { 2 } } } { 3 x ^ { 3 } y ^ { 4 } }$

Question 14

Simplify the radical expression. Assume that all variables represent positive real numbers.
-$\sqrt [ 3 ] { \mathrm { p } ^ { 26 } }$

Accepted Answer

A)  $\mathrm { p } ^ { 8 } \sqrt [ 3 ] { \mathrm { p } ^ { 2 } }$ 
B)  $\sqrt [ 3 ] { \mathrm { p } ^ { 26 } }$ 
C)  $p \sqrt [ 3 ] { \mathrm { p } ^ { 23 } }$ 
D)  $\mathrm { p } ^ { 10 }$ 
A)  $\mathrm { p } ^ { 8 } \sqrt [ 3 ] { \mathrm { p } ^ { 2 } }$ 
B)  $\sqrt [ 3 ] { \mathrm { p } ^ { 26 } }$ 
C)  $p \sqrt [ 3 ] { \mathrm { p } ^ { 23 } }$ 
D)  $\mathrm { p } ^ { 10 }$

Question 15

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

Accepted Answer

A)  35 
B)  $10 \sqrt { 7 }$ 
C)  $25 \sqrt { 7 }$ 
D)  $5 \sqrt { 7 }$ 
A)  35 
B)  $10 \sqrt { 7 }$ 
C)  $25 \sqrt { 7 }$ 
D)  $5 \sqrt { 7 }$

Question 16

Fill in the blank with one of the words or phrases listed below. $$\begin{array} { l l l } 
\text { index rationalizing } & \text { conjugate } & \text { principal square rootcube root midpoint } \
\text { complex numberlike radicals } & \text { radicand } & \text { imaginary unit } \quad \text { distance }
\end{array}$$
-The ________of $\sqrt { 3 } + 2$ is $\sqrt { 3 } - 2$

Accepted Answer

A)  radicand 
B)  imaginary unit 
C)  index 
D)  conjugate 
A)  radicand 
B)  imaginary unit 
C)  index 
D)  conjugate

Question 17

Solve.
-$\sqrt { 43 - x } = x - 1$

Accepted Answer

A)  $- 6$ 
B)  $- 6,7$ 
C)  $\varnothing$ 
D)  7 
A)  $- 6$ 
B)  $- 6,7$ 
C)  $\varnothing$ 
D)  7

Question 18

Add or subtract. Assume all variables represent positive real numbers.
-$\sqrt { 6 a } - 2 \sqrt { 54 a } + 6 \sqrt { 24 a }$

Accepted Answer

A)  $7 \sqrt { 84 a }$ 
B)  $4 \sqrt { 6 a }$ 
C)  $4 \sqrt { 84 a }$ 
D)  $7 \sqrt { 6 a }$ 
A)  $7 \sqrt { 84 a }$ 
B)  $4 \sqrt { 6 a }$ 
C)  $4 \sqrt { 84 a }$ 
D)  $7 \sqrt { 6 a }$

Question 19

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

Accepted Answer

A)  1 
B)  25 
C)  7 
D)  4 
A)  1 
B)  25 
C)  7 
D)  4

Question 20

Use radical notation to write the expression. Simplify if possible.
-$3 x ^ { 3 / 7 }$

Accepted Answer

A)  $\sqrt [ 7 ] { 3 x ^ { 3 } }$ 
B)  $\sqrt [ 7 ] { 27 x ^ { 3 } }$ 
C)  $3 \sqrt [ 7 ] { x ^ { 3 } }$ 
D)  $\sqrt [ 3 ] { 3 x ^ { 7 } }$ 
A)  $\sqrt [ 7 ] { 3 x ^ { 3 } }$ 
B)  $\sqrt [ 7 ] { 27 x ^ { 3 } }$ 
C)  $3 \sqrt [ 7 ] { x ^ { 3 } }$ 
D)  $\sqrt [ 3 ] { 3 x ^ { 7 } }$

Solve. - $\sqrt { 6 x + 3 } = \sqrt { 4 x + 3 } - 1$

Perform the indicated operation. Write the result in the form a + bi. - $\frac { 7 } { 4 + 8 \mathrm { i } }$

Use rational exponents to write as a single radical expression. - $\sqrt [ 5 ] { 2 } \cdot \sqrt [ 3 ] { 3 }$

Find the square root. Assume that all variables represent positive real numbers. - $\sqrt { - 484 }$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { 6 \sqrt { 2 } + \sqrt { 6 } } { 3 \sqrt { 2 } - \sqrt { 6 } }$

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

Use the properties of exponents to simplify the expression. Write with positive exponents. - $x ^ { 1 / 8 } \cdot x ^ { 7 / 8 }$

Use the quotient rule to divide and simplify. - $\sqrt { \frac { 32 x ^ { 2 } y } { 25 } }$

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

Find the distance between the pair of points. - $( 2 , - 5 )$ and $( 6 , - 7 )$

Write with positive exponents. Simplify if possible. - $16 ^ { - 5 / 4 }$

Solve. - $\sqrt { x + 3 } = 8$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { 8 x } { \sqrt [ 5 ] { 27 x ^ { 17 } y ^ { 23 } } }$

Simplify the radical expression. Assume that all variables represent positive real numbers. - $\sqrt [ 3 ] { \mathrm { p } ^ { 26 } }$

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

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 -The ________of $\sqrt { 3 } + 2$ is $\sqrt { 3 } - 2$

Solve. - $\sqrt { 43 - x } = x - 1$

Add or subtract. Assume all variables represent positive real numbers. - $\sqrt { 6 a } - 2 \sqrt { 54 a } + 6 \sqrt { 24 a }$

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

Use radical notation to write the expression. Simplify if possible. - $3 x ^ { 3 / 7 }$

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

Solve. - 6x+3=4x+3−1\sqrt { 6 x + 3 } = \sqrt { 4 x + 3 } - 16x+3​=4x+3​−1

Perform the indicated operation. Write the result in the form a + bi. - 74+8i\frac { 7 } { 4 + 8 \mathrm { i } }4+8i7​

Use rational exponents to write as a single radical expression. - 25⋅33\sqrt [ 5 ] { 2 } \cdot \sqrt [ 3 ] { 3 }52​⋅33​

Find the square root. Assume that all variables represent positive real numbers. - −484\sqrt { - 484 }−484​

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - 62+632−6\frac { 6 \sqrt { 2 } + \sqrt { 6 } } { 3 \sqrt { 2 } - \sqrt { 6 } }32​−6​62​+6​​

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - (5+33)(5−33)( 5 + \sqrt [ 3 ] { 3 } ) ( 5 - \sqrt [ 3 ] { 3 } )(5+33​)(5−33​)

Use the properties of exponents to simplify the expression. Write with positive exponents. - x1/8⋅x7/8x ^ { 1 / 8 } \cdot x ^ { 7 / 8 }x1/8⋅x7/8

Use the quotient rule to divide and simplify. - 32x2y25\sqrt { \frac { 32 x ^ { 2 } y } { 25 } }2532x2y​​

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

Find the distance between the pair of points. - (2,−5)( 2 , - 5 )(2,−5) and (6,−7)( 6 , - 7 )(6,−7)

Write with positive exponents. Simplify if possible. - 16−5/416 ^ { - 5 / 4 }16−5/4

Solve. - x+3=8\sqrt { x + 3 } = 8x+3​=8

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - 8x27x17y235\frac { 8 x } { \sqrt [ 5 ] { 27 x ^ { 17 } y ^ { 23 } } }527x17y23​8x​

Simplify the radical expression. Assume that all variables represent positive real numbers. - p263\sqrt [ 3 ] { \mathrm { p } ^ { 26 } }3p26​

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

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 -The ________of 3+2\sqrt { 3 } + 23​+2 is 3−2\sqrt { 3 } - 23​−2

Solve. - 43−x=x−1\sqrt { 43 - x } = x - 143−x​=x−1

Add or subtract. Assume all variables represent positive real numbers. - 6a−254a+624a\sqrt { 6 a } - 2 \sqrt { 54 a } + 6 \sqrt { 24 a }6a​−254a​+624a​

Solve. - x+3−2=0\sqrt { x + 3 } - 2 = 0x+3​−2=0

Use radical notation to write the expression. Simplify if possible. - 3x3/73 x ^ { 3 / 7 }3x3/7

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

Solve. - $\sqrt { 6 x + 3 } = \sqrt { 4 x + 3 } - 1$

Perform the indicated operation. Write the result in the form a + bi. - $\frac { 7 } { 4 + 8 \mathrm { i } }$

Use rational exponents to write as a single radical expression. - $\sqrt [ 5 ] { 2 } \cdot \sqrt [ 3 ] { 3 }$

Find the square root. Assume that all variables represent positive real numbers. - $\sqrt { - 484 }$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { 6 \sqrt { 2 } + \sqrt { 6 } } { 3 \sqrt { 2 } - \sqrt { 6 } }$

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

Use the properties of exponents to simplify the expression. Write with positive exponents. - $x ^ { 1 / 8 } \cdot x ^ { 7 / 8 }$

Use the quotient rule to divide and simplify. - $\sqrt { \frac { 32 x ^ { 2 } y } { 25 } }$

Find the distance between the pair of points. - $( 2 , - 5 )$ and $( 6 , - 7 )$

Write with positive exponents. Simplify if possible. - $16 ^ { - 5 / 4 }$

Solve. - $\sqrt { x + 3 } = 8$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { 8 x } { \sqrt [ 5 ] { 27 x ^ { 17 } y ^ { 23 } } }$

Simplify the radical expression. Assume that all variables represent positive real numbers. - $\sqrt [ 3 ] { \mathrm { p } ^ { 26 } }$

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 -The ________of $\sqrt { 3 } + 2$ is $\sqrt { 3 } - 2$

Solve. - $\sqrt { 43 - x } = x - 1$

Add or subtract. Assume all variables represent positive real numbers. - $\sqrt { 6 a } - 2 \sqrt { 54 a } + 6 \sqrt { 24 a }$

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

Use radical notation to write the expression. Simplify if possible. - $3 x ^ { 3 / 7 }$