Question 1

Simplify the radical expression. Assume that all variables represent positive real numbers.
-$\sqrt { 28 }$

Accepted Answer

A)  $2 \sqrt { 7 }$ 
B)  $\sqrt { 28 }$ 
C)  $4 \sqrt { 7 }$ 
D)  14 
A)  $2 \sqrt { 7 }$ 
B)  $\sqrt { 28 }$ 
C)  $4 \sqrt { 7 }$ 
D)  14

Question 2

Find the midpoint of the line segment whose endpoints are given.
-$( 8 , - 2 ) , ( - 1,1 )$

Accepted Answer

A)  $( 9 , - 3 )$ 
B)  $( 7 , - 1 )$ 
C)  $\left( \frac { 7 } { 2 } , - \frac { 1 } { 2 } \right)$ 
D)  $\left( \frac { 9 } { 2 } , - \frac { 3 } { 2 } \right)$ 
A)  $( 9 , - 3 )$ 
B)  $( 7 , - 1 )$ 
C)  $\left( \frac { 7 } { 2 } , - \frac { 1 } { 2 } \right)$ 
D)  $\left( \frac { 9 } { 2 } , - \frac { 3 } { 2 } \right)$

Question 3

Perform the indicated operation and simplify. Write the result in the form a $a + bi$
-$( 9 - 6 i ) ^ { 2 }$

Accepted Answer

A)  $117 - 108 \mathrm { i }$ 
B)  $81 - 108 i + 36 i ^ { 2 }$ 
C)  $45 + 0 \mathrm { i }$ 
D)  $45 - 108 \mathrm { i }$ 
A)  $117 - 108 \mathrm { i }$ 
B)  $81 - 108 i + 36 i ^ { 2 }$ 
C)  $45 + 0 \mathrm { i }$ 
D)  $45 - 108 \mathrm { i }$

Question 4

Find the cube root.
-$\sqrt [ 3 ] { 729 x ^ { 6 } }$

Accepted Answer

A)  $9 x ^ { 2 }$ 
B)  not a real number 
C)  $81 x ^ { 2 }$ 
D)  $27 x ^ { 2 }$ 
A)  $9 x ^ { 2 }$ 
B)  not a real number 
C)  $81 x ^ { 2 }$ 
D)  $27 x ^ { 2 }$

Question 5

Perform the indicated operation. Write the result in the form a + bi.
-$\frac { 7 } { 9 i }$

Accepted Answer

A)  $- \frac { 7 } { 9 } + 0 i$ 
B)  $0 + \frac { 7 } { 9 } \mathrm { i }$ 
C)  $\frac { 7 } { 9 } + 0 \mathrm { i }$ 
D)  $0 - \frac { 7 } { 9 } \mathrm { i }$ 
A)  $- \frac { 7 } { 9 } + 0 i$ 
B)  $0 + \frac { 7 } { 9 } \mathrm { i }$ 
C)  $\frac { 7 } { 9 } + 0 \mathrm { i }$ 
D)  $0 - \frac { 7 } { 9 } \mathrm { i }$

Question 6

Use a calculator to approximate the square root to 3 decimal places. Check to see that the approximation is reasonable.
-$\sqrt { 8 }$

Accepted Answer

A)  $2.833$ 
B)  $8.000$ 
C)  $2.828$ 
D)  $2.825$ 
A)  $2.833$ 
B)  $8.000$ 
C)  $2.828$ 
D)  $2.825$

Question 7

Simplify the radical expression. Assume that all variables represent positive real numbers.
-$\sqrt [ 3 ] { 343 x ^ { 4 } y ^ { 5 } }$

Accepted Answer

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

Question 8

Solve.
-$\sqrt { 6 x - 14 } = 5 - x$

Accepted Answer

A)  3 
B)  $\varnothing$ 
C)  3,13 
D)  13 
A)  3 
B)  $\varnothing$ 
C)  3,13 
D)  13

Question 9

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

Accepted Answer

A)  $\frac { 12 + 2 \sqrt { 3 } } { - 3 }$ 
B)  $\frac { 12 - 2 \sqrt { 3 } } { 33 }$ 
C)  $\frac { 2 } { 6 } - \frac { 2 } { \sqrt { 3 } }$ 
D)  $\frac { 12 + 2 \sqrt { 3 } } { 33 }$ 
A)  $\frac { 12 + 2 \sqrt { 3 } } { - 3 }$ 
B)  $\frac { 12 - 2 \sqrt { 3 } } { 33 }$ 
C)  $\frac { 2 } { 6 } - \frac { 2 } { \sqrt { 3 } }$ 
D)  $\frac { 12 + 2 \sqrt { 3 } } { 33 }$

Question 10

Use the quotient rule to divide and simplify.
-$\sqrt [ 5 ] { \frac { x ^ { 4 } } { 729 } }$

Accepted Answer

A)  $\frac { x \sqrt [ 4 ] { x } } { 3 \sqrt [ 4 ] { 9 } }$ 
B)  $\frac { \sqrt [ 5 ] { x ^ { 4 } } } { 3 \sqrt [ 5 ] { 3 } }$ 
C)  $\frac { \sqrt [ 5 ] { x ^ { 4 } } } { 3 }$ 
D)  $\frac { x \sqrt [ 4 ] { x } } { 3 }$ 
A)  $\frac { x \sqrt [ 4 ] { x } } { 3 \sqrt [ 4 ] { 9 } }$ 
B)  $\frac { \sqrt [ 5 ] { x ^ { 4 } } } { 3 \sqrt [ 5 ] { 3 } }$ 
C)  $\frac { \sqrt [ 5 ] { x ^ { 4 } } } { 3 }$ 
D)  $\frac { x \sqrt [ 4 ] { x } } { 3 }$

Question 11

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\frac { 11 } { \sqrt [ 3 ] { 2 } }$

Accepted Answer

A)  $\frac { 11 \sqrt { 2 } } { 2 }$ 
B)  $\frac { 11 \sqrt [ 3 ] { 2 } } { 2 }$ 
C)  $\frac { 11 \sqrt [ 3 ] { 4 } } { 2 }$ 
D)  $\frac { \sqrt [ 3 ] { 22 } } { 2 }$ 
A)  $\frac { 11 \sqrt { 2 } } { 2 }$ 
B)  $\frac { 11 \sqrt [ 3 ] { 2 } } { 2 }$ 
C)  $\frac { 11 \sqrt [ 3 ] { 4 } } { 2 }$ 
D)  $\frac { \sqrt [ 3 ] { 22 } } { 2 }$

Question 12

Simplify the radical expression. Assume that all variables represent positive real numbers.
-$\sqrt [ 5 ] { 243 \mathrm { x } ^ { 3 } \mathrm { y } ^ { 22 } }$

Accepted Answer

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

Question 13

Find the distance between the pair of points.
-$( - 4 \sqrt { 11 } , - 2 )$ and $( - 3 \sqrt { 11 } , 3 )$

Accepted Answer

A)  5 units 
B)  36 units 
C)  $\sqrt { 6 }$ units 
D)  6 units 
A)  5 units 
B)  36 units 
C)  $\sqrt { 6 }$ units 
D)  6 units

Question 14

Raise to the power or find the root. Assume that all variables represent positive numbers. Write with only positive
exponents.
-$\sqrt [ 3 ] { - 27 a ^ { 9 } b ^ { 3 } }$

Accepted Answer

A)  $3 a ^ { 12 } b ^ { 6 }$ 
B)  $27 a ^ { 3 } b$ 
C)  $- 3 a ^ { 3 } b$ 
D)  $3 a ^ { 3 } b$ 
A)  $3 a ^ { 12 } b ^ { 6 }$ 
B)  $27 a ^ { 3 } b$ 
C)  $- 3 a ^ { 3 } b$ 
D)  $3 a ^ { 3 } b$

Question 15

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\frac { 7 \sqrt { 3 } } { \sqrt { 5 } }$

Accepted Answer

A)  $\frac { \sqrt { 735 } } { 5 }$ 
B)  $\frac { 7 \sqrt { 15 } } { 5 }$ 
C)  $\frac { 35 \sqrt { 5 } } { 3 }$ 
D)  $\frac { \sqrt { 15 } } { 35 }$ 
A)  $\frac { \sqrt { 735 } } { 5 }$ 
B)  $\frac { 7 \sqrt { 15 } } { 5 }$ 
C)  $\frac { 35 \sqrt { 5 } } { 3 }$ 
D)  $\frac { \sqrt { 15 } } { 35 }$

Question 16

Raise to the power or find the root. Assume that all variables represent positive numbers. Write with only positive
exponents.
-$\left( \frac { 1 } { 16 } \right) ^ { - 1 / 4 }$

Accepted Answer

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

Question 17

Multiply or divide.
-$\frac { \sqrt { 25 } } { \sqrt { - 3 } }$

Accepted Answer

A)  $\frac { 5 i \sqrt { 3 } } { 3 }$ 
B)  $\frac { - 5 \sqrt { 3 } } { 3 }$ 
C)  $\frac { - 5 i \sqrt { 3 } } { 3 }$ 
D)  $\frac { 5 \sqrt { 3 } } { 3 }$ 
A)  $\frac { 5 i \sqrt { 3 } } { 3 }$ 
B)  $\frac { - 5 \sqrt { 3 } } { 3 }$ 
C)  $\frac { - 5 i \sqrt { 3 } } { 3 }$ 
D)  $\frac { 5 \sqrt { 3 } } { 3 }$

Question 18

Use rational exponents to write as a single radical expression.
-$\frac { \sqrt [ 5 ] { \mathrm { y } ^ { 2 } } } { \sqrt [ 6 ] { \mathrm { y } } }$

Accepted Answer

A)  $\sqrt [ 30 ] { \mathrm { y } }$ 
B)  $\sqrt [ 5 ] { \mathrm { y } }$ 
C)  $\sqrt { \mathrm { y } }$ 
D)  $\sqrt [ 30 ] { \mathrm { y } ^ { 7 } }$ 
A)  $\sqrt [ 30 ] { \mathrm { y } }$ 
B)  $\sqrt [ 5 ] { \mathrm { y } }$ 
C)  $\sqrt { \mathrm { y } }$ 
D)  $\sqrt [ 30 ] { \mathrm { y } ^ { 7 } }$

Question 19

Use radical notation to write the expression. Simplify if possible.
-$( 2 x + 1 ) ^ { 3 / 5 }$

Accepted Answer

A)  $\sqrt [ 5 ] { 8 x ^ { 3 } + 1 }$ 
B)  $\sqrt [ 5 ] { 2 x ^ { 3 } + 1 }$ 
C)  $\sqrt [ 3 ] { ( 2 x + 1 ) ^ { 5 } }$ 
D)  $\sqrt [ 5 ] { ( 2 x + 1 ) ^ { 3 } }$ 
A)  $\sqrt [ 5 ] { 8 x ^ { 3 } + 1 }$ 
B)  $\sqrt [ 5 ] { 2 x ^ { 3 } + 1 }$ 
C)  $\sqrt [ 3 ] { ( 2 x + 1 ) ^ { 5 } }$ 
D)  $\sqrt [ 5 ] { ( 2 x + 1 ) ^ { 3 } }$

Question 20

Use the properties of exponents to simplify the expression. Write with positive exponents.
-$y ^ { 5 / 9 } \left( y ^ { 2 / 9 } - 4 y ^ { 3 / 9 } \right)$

Accepted Answer

A)  $y ^ { 3 / 9 } - 4 y ^ { 2 / 9 }$ 
B)  $\mathrm { y }^{ 10 / 81} - 4 \mathrm { y } ^{15 / 81}$ 
C)  $y ^ { 5 / 2 } - 4 y ^ { 5 / 3 }$ 
D)  $y ^ { 7 / 9 } - 4 y ^ { 8 / 9 }$ 
A)  $y ^ { 3 / 9 } - 4 y ^ { 2 / 9 }$ 
B)  $\mathrm { y }^{ 10 / 81} - 4 \mathrm { y } ^{15 / 81}$ 
C)  $y ^ { 5 / 2 } - 4 y ^ { 5 / 3 }$ 
D)  $y ^ { 7 / 9 } - 4 y ^ { 8 / 9 }$

Simplify the radical expression. Assume that all variables represent positive real numbers. - $\sqrt { 28 }$

Find the midpoint of the line segment whose endpoints are given. - $( 8 , - 2 ) , ( - 1,1 )$

Perform the indicated operation and simplify. Write the result in the form a $a + bi$ - $( 9 - 6 i ) ^ { 2 }$

Find the cube root. - $\sqrt [ 3 ] { 729 x ^ { 6 } }$

Perform the indicated operation. Write the result in the form a + bi. - $\frac { 7 } { 9 i }$

Use a calculator to approximate the square root to 3 decimal places. Check to see that the approximation is reasonable. - $\sqrt { 8 }$

Simplify the radical expression. Assume that all variables represent positive real numbers. - $\sqrt [ 3 ] { 343 x ^ { 4 } y ^ { 5 } }$

Solve. - $\sqrt { 6 x - 14 } = 5 - x$

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

Use the quotient rule to divide and simplify. - $\sqrt [ 5 ] { \frac { x ^ { 4 } } { 729 } }$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { 11 } { \sqrt [ 3 ] { 2 } }$

Simplify the radical expression. Assume that all variables represent positive real numbers. - $\sqrt [ 5 ] { 243 \mathrm { x } ^ { 3 } \mathrm { y } ^ { 22 } }$

Find the distance between the pair of points. - $( - 4 \sqrt { 11 } , - 2 )$ and $( - 3 \sqrt { 11 } , 3 )$

Raise to the power or find the root. Assume that all variables represent positive numbers. Write with only positive exponents. - $\sqrt [ 3 ] { - 27 a ^ { 9 } b ^ { 3 } }$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { 7 \sqrt { 3 } } { \sqrt { 5 } }$

Raise to the power or find the root. Assume that all variables represent positive numbers. Write with only positive exponents. - $\left( \frac { 1 } { 16 } \right) ^ { - 1 / 4 }$

Multiply or divide. - $\frac { \sqrt { 25 } } { \sqrt { - 3 } }$

Use rational exponents to write as a single radical expression. - $\frac { \sqrt [ 5 ] { \mathrm { y } ^ { 2 } } } { \sqrt [ 6 ] { \mathrm { y } } }$

Use radical notation to write the expression. Simplify if possible. - $( 2 x + 1 ) ^ { 3 / 5 }$

Use the properties of exponents to simplify the expression. Write with positive exponents. - $y ^ { 5 / 9 } \left( y ^ { 2 / 9 } - 4 y ^ { 3 / 9 } \right)$

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

Simplify the radical expression. Assume that all variables represent positive real numbers. - 28\sqrt { 28 }28​

Find the midpoint of the line segment whose endpoints are given. - (8,−2),(−1,1)( 8 , - 2 ) , ( - 1,1 )(8,−2),(−1,1)

Perform the indicated operation and simplify. Write the result in the form a a+bia + bia+bi - (9−6i)2( 9 - 6 i ) ^ { 2 }(9−6i)2

Find the cube root. - 729x63\sqrt [ 3 ] { 729 x ^ { 6 } }3729x6​

Perform the indicated operation. Write the result in the form a + bi. - 79i\frac { 7 } { 9 i }9i7​

Use a calculator to approximate the square root to 3 decimal places. Check to see that the approximation is reasonable. - 8\sqrt { 8 }8​

Simplify the radical expression. Assume that all variables represent positive real numbers. - 343x4y53\sqrt [ 3 ] { 343 x ^ { 4 } y ^ { 5 } }3343x4y5​

Solve. - 6x−14=5−x\sqrt { 6 x - 14 } = 5 - x6x−14​=5−x

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - 26−3\frac { 2 } { 6 - \sqrt { 3 } }6−3​2​

Use the quotient rule to divide and simplify. - x47295\sqrt [ 5 ] { \frac { x ^ { 4 } } { 729 } }5729x4​​

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - 1123\frac { 11 } { \sqrt [ 3 ] { 2 } }32​11​

Simplify the radical expression. Assume that all variables represent positive real numbers. - 243x3y225\sqrt [ 5 ] { 243 \mathrm { x } ^ { 3 } \mathrm { y } ^ { 22 } }5243x3y22​

Find the distance between the pair of points. - (−411,−2)( - 4 \sqrt { 11 } , - 2 )(−411​,−2) and (−311,3)( - 3 \sqrt { 11 } , 3 )(−311​,3)

Raise to the power or find the root. Assume that all variables represent positive numbers. Write with only positive exponents. - −27a9b33\sqrt [ 3 ] { - 27 a ^ { 9 } b ^ { 3 } }3−27a9b3​

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - 735\frac { 7 \sqrt { 3 } } { \sqrt { 5 } }5​73​​

Raise to the power or find the root. Assume that all variables represent positive numbers. Write with only positive exponents. - (116)−1/4\left( \frac { 1 } { 16 } \right) ^ { - 1 / 4 }(161​)−1/4

Multiply or divide. - 25−3\frac { \sqrt { 25 } } { \sqrt { - 3 } }−3​25​​

Use rational exponents to write as a single radical expression. - y25y6\frac { \sqrt [ 5 ] { \mathrm { y } ^ { 2 } } } { \sqrt [ 6 ] { \mathrm { y } } }6y​5y2​​

Use radical notation to write the expression. Simplify if possible. - (2x+1)3/5( 2 x + 1 ) ^ { 3 / 5 }(2x+1)3/5

Use the properties of exponents to simplify the expression. Write with positive exponents. - y5/9(y2/9−4y3/9)y ^ { 5 / 9 } \left( y ^ { 2 / 9 } - 4 y ^ { 3 / 9 } \right)y5/9(y2/9−4y3/9)