Question 1

Simplify by rationalizing the denominator.
-$\frac { 5 } { \sqrt { y + 3 } - 4 }$

Accepted Answer

A)  $\frac { 5 \sqrt { y + 3 } - 20 } { y + 13 }$ 
B)  $\frac { 5 \sqrt { y + 3 } + 20 } { y - 13 }$ 
C)  $\frac { 5 } { y - 13 }$ 
D)  $\frac { 5 \sqrt { y - 3 } + 20 } { y - 19 }$ 
A)  $\frac { 5 \sqrt { y + 3 } - 20 } { y + 13 }$ 
B)  $\frac { 5 \sqrt { y + 3 } + 20 } { y - 13 }$ 
C)  $\frac { 5 } { y - 13 }$ 
D)  $\frac { 5 \sqrt { y - 3 } + 20 } { y - 19 }$

Question 2

Write the expression in exponential form. Assume that all variables represent positive real numbers.
-$\sqrt [ 3 ] { x ^ { 13 } }$

Accepted Answer

A)  $x ^ { 9 / 13 }$ 
B)  $x ^ { - 13 / 9 }$ 
C)  $x ^ { 13 / 9 }$ 
D)  $x ^ { 4 }$ 
A)  $x ^ { 9 / 13 }$ 
B)  $x ^ { - 13 / 9 }$ 
C)  $x ^ { 13 / 9 }$ 
D)  $x ^ { 4 }$

Question 3

Multiply. Write the result in the form a + bi.
-( $( 3 + i \sqrt { 3 } ) ( 3 - i \sqrt { 3 } )$

Accepted Answer

A) 18 
B) 12 
C) 6 
D)  $9 + 3 \mathrm { i }$ 
A) 18 
B) 12 
C) 6 
D)  $9 + 3 \mathrm { i }$

Question 4

Solve and check your solution(s). If the equation has no real solution, so state.
-$x - \sqrt { 3 x - 2 } = 4$

Accepted Answer

A) 2, 9 
B) 1, 2 
C) -1 
D) 9 
A) 2, 9 
B) 1, 2 
C) -1 
D) 9

Question 5

Simplify.
-$\frac { \sqrt [ 3 ] { 5 } } { \sqrt [ 3 ] { 1080 } }$

Accepted Answer

A)  $\frac { 6 } { \sqrt [ 3 ] { 5 } }$ 
B)  $\frac { 1 } { 6 }$ 
C) 6 
D)  $\frac { \sqrt [ 3 ] { 5 } } { 6 }$ 
A)  $\frac { 6 } { \sqrt [ 3 ] { 5 } }$ 
B)  $\frac { 1 } { 6 }$ 
C) 6 
D)  $\frac { \sqrt [ 3 ] { 5 } } { 6 }$

Question 6

Write the expression as a complex number in the form a + bi.
-$\sqrt { 36 }$

Accepted Answer

A) 0 + 6i 
B) 6 + 0i 
C) -6 + 0i 
D) 0 - 6i 
A) 0 + 6i 
B) 6 + 0i 
C) -6 + 0i 
D) 0 - 6i

Question 7

Simplify the radical expression. Write the answer in radical form when appropriate.
-$\sqrt [ 20 ] { x ^ { 4 } }$

Accepted Answer

A)  $\sqrt [ 5 ] { \frac { 1 } { x } }$ 
B)  $\sqrt [ 4 ] { x }$ 
C)  $\sqrt [ 16 ] { x }$ 
D)  $\sqrt [ 5 ] { x }$ 
A)  $\sqrt [ 5 ] { \frac { 1 } { x } }$ 
B)  $\sqrt [ 4 ] { x }$ 
C)  $\sqrt [ 16 ] { x }$ 
D)  $\sqrt [ 5 ] { x }$

Question 8

Simplify the radical expression. Write the answer in radical form when appropriate.
-$\left( \sqrt [ 6 ] { x ^ { 5 } y z ^ { 5 } } \right) ^ { 3 }$

Accepted Answer

A)  $x ^ { 10 } y ^ { 2 } z ^ { 10 }$ 
B)  $\sqrt { x ^ { 5 } y z ^ { 5 } }$ 
C)  $\sqrt { x ^ { 15 } y ^ { 3 } z ^ { 15 } }$ 
D)  $x ^ { 5 } y ^ { 2 } z ^ { 5 }$ 
A)  $x ^ { 10 } y ^ { 2 } z ^ { 10 }$ 
B)  $\sqrt { x ^ { 5 } y z ^ { 5 } }$ 
C)  $\sqrt { x ^ { 15 } y ^ { 3 } z ^ { 15 } }$ 
D)  $x ^ { 5 } y ^ { 2 } z ^ { 5 }$

Question 9

Simplify.
-$\frac { \sqrt { 50 } } { \sqrt { 2 } }$

Accepted Answer

A) 5 
B)  $5 \sqrt { 2 }$ 
C)  $\frac { 5 } { \sqrt { 2 } }$ 
D) 2 
A) 5 
B)  $5 \sqrt { 2 }$ 
C)  $\frac { 5 } { \sqrt { 2 } }$ 
D) 2

Question 10

Simplify. Write the answer in exponential form without negative exponents.
-$\left( x ^ { 1 / 3 } \right) ^ { - 3 }$

Accepted Answer

A) x 
B)  $\frac { 1 } { x }$ 
C)  $\mathrm { x } ^ { 1 / 9 }$ 
D)  $\frac { 1 } { x ^ { 9 } }$ 
A) x 
B)  $\frac { 1 } { x }$ 
C)  $\mathrm { x } ^ { 1 / 9 }$ 
D)  $\frac { 1 } { x ^ { 9 } }$

Question 11

Write the expression as a complex number in the form a + bi.
-$\sqrt { - 2 } - \sqrt { - 169 }$

Accepted Answer

A)  $0 + ( \sqrt { 2 } + 13 ) \mathrm { i }$ 
B)  $13 + \sqrt { 2 } i$ 
C)  $0 + ( \sqrt { 2 } - 13 ) \mathrm { i }$ 
D)  $- 13 + \sqrt { 2 } i$ 
A)  $0 + ( \sqrt { 2 } + 13 ) \mathrm { i }$ 
B)  $13 + \sqrt { 2 } i$ 
C)  $0 + ( \sqrt { 2 } - 13 ) \mathrm { i }$ 
D)  $- 13 + \sqrt { 2 } i$

Question 12

Evaluate if possible. If the expression is not a real number, so state.
-$16 ^ { 1 / 4 }$

Accepted Answer

A) 16 
B) 8 
C) 2 
D) 32 
A) 16 
B) 8 
C) 2 
D) 32

Question 13

Evaluate the radical expression if it is a real number. Use a calculator to approximate irrational numbers to the nearest
hundredth. If the expression is not a real number, so state.
-$\sqrt { \frac { 36 } { 169 } }$

Accepted Answer

A)  $\frac { 3 } { 7 }$ 
B)  $\frac { 6 } { 13 }$ 
C)  $\frac { 3 } { 14 }$ 
D) 137 
A)  $\frac { 3 } { 7 }$ 
B)  $\frac { 6 } { 13 }$ 
C)  $\frac { 3 } { 14 }$ 
D) 137

Question 14

Evaluate the radical expression if it is a real number. Use a calculator to approximate irrational numbers to the nearest
hundredth. If the expression is not a real number, so state.
-$$\sqrt [ 5 ] { 243 }$$

Accepted Answer

A) 243 
B) 3 
C) -3 
D) not a real number 
A) 243 
B) 3 
C) -3 
D) not a real number

Question 15

Write the expression as a complex number in the form a + bi.
-$10 + \sqrt { 108 }$

Accepted Answer

A) 18 + 0i 
B)  $10 + 6 \sqrt { 3 } \mathrm { i }$ 
C)  $( 10 + 6 \sqrt { 3 } ) + 0 \mathrm { i }$ 
D)  $10 + 3 \sqrt { 6 } i$ 
A) 18 + 0i 
B)  $10 + 6 \sqrt { 3 } \mathrm { i }$ 
C)  $( 10 + 6 \sqrt { 3 } ) + 0 \mathrm { i }$ 
D)  $10 + 3 \sqrt { 6 } i$

Question 16

Evaluate the radical expression if it is a real number. Use a calculator to approximate irrational numbers to the nearest
hundredth. If the expression is not a real number, so state.
-$$- \sqrt [ 4 ] { 81 }$$

Accepted Answer

A) -36 
B) -12 
C) -3 
D) not a real number 
A) -36 
B) -12 
C) -3 
D) not a real number

Question 17

Solve the problem.
-Abigail needs to determine the distance at certain points across a lake. Her crew and she are able to measure the distances shown on the diagram below. Find how wide the lake is to the nearest tenth of a meter.

Accepted Answer

A) 70 m 
B) 115.3 m 
C) 8.4 m 
D) 143.2 m 
A) 70 m 
B) 115.3 m 
C) 8.4 m 
D) 143.2 m

Question 18

Solve. You will have to square both sides of the equation twice to eliminate all radicals.
-$\sqrt { x + 164 } - \sqrt { x + 59 } = 5$

Accepted Answer

A) -5 
B) 6 
C) 5 
D) no real solution 
A) -5 
B) 6 
C) 5 
D) no real solution

Question 19

Evaluate if possible. If the expression is not a real number, so state.
-$110 ^ { - 1 / 2 }$

Accepted Answer

A)  $\frac { 1 } { 100 }$ 
B)  $- \frac { 1 } { 10 }$ 
C)  $\frac { 1 } { 10 }$ 
D) - 10 
A)  $\frac { 1 } { 100 }$ 
B)  $- \frac { 1 } { 10 }$ 
C)  $\frac { 1 } { 10 }$ 
D) - 10

Question 20

Divide.
-$\frac { 1 + i } { 1 - i }$

Accepted Answer

A) 2 + 2i 
B) -2 + 2i 
C) Undefined 
D) i 
A) 2 + 2i 
B) -2 + 2i 
C) Undefined 
D) i

Simplify by rationalizing the denominator. - $\frac { 5 } { \sqrt { y + 3 } - 4 }$

Write the expression in exponential form. Assume that all variables represent positive real numbers. - $\sqrt [ 3 ] { x ^ { 13 } }$

Multiply. Write the result in the form a + bi. -( $( 3 + i \sqrt { 3 } ) ( 3 - i \sqrt { 3 } )$

Solve and check your solution(s). If the equation has no real solution, so state. - $x - \sqrt { 3 x - 2 } = 4$

Simplify. - $\frac { \sqrt [ 3 ] { 5 } } { \sqrt [ 3 ] { 1080 } }$

Write the expression as a complex number in the form a + bi. - $\sqrt { 36 }$

Simplify the radical expression. Write the answer in radical form when appropriate. - $\sqrt [ 20 ] { x ^ { 4 } }$

Simplify the radical expression. Write the answer in radical form when appropriate. - $\left( \sqrt [ 6 ] { x ^ { 5 } y z ^ { 5 } } \right) ^ { 3 }$

Simplify. - $\frac { \sqrt { 50 } } { \sqrt { 2 } }$

Simplify. Write the answer in exponential form without negative exponents. - $\left( x ^ { 1 / 3 } \right) ^ { - 3 }$

Write the expression as a complex number in the form a + bi. - $\sqrt { - 2 } - \sqrt { - 169 }$

Evaluate if possible. If the expression is not a real number, so state. - $16 ^ { 1 / 4 }$

Evaluate the radical expression if it is a real number. Use a calculator to approximate irrational numbers to the nearest hundredth. If the expression is not a real number, so state. - $\sqrt { \frac { 36 } { 169 } }$

Evaluate the radical expression if it is a real number. Use a calculator to approximate irrational numbers to the nearest hundredth. If the expression is not a real number, so state. - $\sqrt [ 5 ] { 243 }$

Write the expression as a complex number in the form a + bi. - $10 + \sqrt { 108 }$

Evaluate the radical expression if it is a real number. Use a calculator to approximate irrational numbers to the nearest hundredth. If the expression is not a real number, so state. - $- \sqrt [ 4 ] { 81 }$

Solve the problem. -Abigail needs to determine the distance at certain points across a lake. Her crew and she are able to measure the distances shown on the diagram below. Find how wide the lake is to the nearest tenth of a meter.

Solve. You will have to square both sides of the equation twice to eliminate all radicals. - $\sqrt { x + 164 } - \sqrt { x + 59 } = 5$

Evaluate if possible. If the expression is not a real number, so state. - $110 ^ { - 1 / 2 }$

Divide. - $\frac { 1 + i } { 1 - i }$

Basic Concepts

Equations and Inequalities

Graphs and Functions

Systems of Equations and Inequalities

Polynomials and Polynomial Functions

Rational Expressions and Equations

Quadratic Functions

Exponential and Logarithmic Functions

Characteristics of Functions and Their Graphs

Conic Sections

Sequences, Series, and Probability

Filters

Exam 7: Roots, Radicals, and Complex Numbers

Simplify by rationalizing the denominator. - 5y+3−4\frac { 5 } { \sqrt { y + 3 } - 4 }y+3​−45​

Write the expression in exponential form. Assume that all variables represent positive real numbers. - x133\sqrt [ 3 ] { x ^ { 13 } }3x13​

Multiply. Write the result in the form a + bi. -( (3+i3)(3−i3)( 3 + i \sqrt { 3 } ) ( 3 - i \sqrt { 3 } )(3+i3​)(3−i3​)

Solve and check your solution(s). If the equation has no real solution, so state. - x−3x−2=4x - \sqrt { 3 x - 2 } = 4x−3x−2​=4

Simplify. - 5310803\frac { \sqrt [ 3 ] { 5 } } { \sqrt [ 3 ] { 1080 } }31080​35​​

Write the expression as a complex number in the form a + bi. - 36\sqrt { 36 }36​

Simplify the radical expression. Write the answer in radical form when appropriate. - x420\sqrt [ 20 ] { x ^ { 4 } }20x4​

Simplify the radical expression. Write the answer in radical form when appropriate. - (x5yz56)3\left( \sqrt [ 6 ] { x ^ { 5 } y z ^ { 5 } } \right) ^ { 3 }(6x5yz5​)3

Simplify. - 502\frac { \sqrt { 50 } } { \sqrt { 2 } }2​50​​

Simplify. Write the answer in exponential form without negative exponents. - (x1/3)−3\left( x ^ { 1 / 3 } \right) ^ { - 3 }(x1/3)−3

Write the expression as a complex number in the form a + bi. - −2−−169\sqrt { - 2 } - \sqrt { - 169 }−2​−−169​

Evaluate if possible. If the expression is not a real number, so state. - 161/416 ^ { 1 / 4 }161/4

Evaluate the radical expression if it is a real number. Use a calculator to approximate irrational numbers to the nearest hundredth. If the expression is not a real number, so state. - 36169\sqrt { \frac { 36 } { 169 } }16936​​

Evaluate the radical expression if it is a real number. Use a calculator to approximate irrational numbers to the nearest hundredth. If the expression is not a real number, so state. - 2435\sqrt [ 5 ] { 243 }5243​

Write the expression as a complex number in the form a + bi. - 10+10810 + \sqrt { 108 }10+108​

Evaluate the radical expression if it is a real number. Use a calculator to approximate irrational numbers to the nearest hundredth. If the expression is not a real number, so state. - −814- \sqrt [ 4 ] { 81 }−481​