Deck 1: A Review of Basic Algebra

Question 1

Simplify the expression. $$\frac { x ^ { 4 } x ^ { 2 } } { x ^ { 3 } x }$$ Write the answer without using negative exponents.Assume that the variable is restricted to those numbers for which the expression is defined. a. $x ^ { 4 }$
b. $x ^ { 3 }$
C. $\quad x ^ { 2 }$
d. $x ^ { 10 }$

Answer 1

C

Question 2

Factor the expression completely. $$4 z ^ { 2 } + 28 z + 49$$
a. $\quad ( 2 z + 7 ) ^ { 2 }$
b. $\quad 7 ( 2 z + 7 )$
c. $\quad ( 2 z + 7 ) ( 2 z - 7 )$
d. $\quad ( 2 z - 7 ) ^ { 2 }$

Answer 2

A

Question 3

Simplify the expression. $$\frac { 1 } { x ^ { - 7 } }$$ Write the answer without using negative exponents.Assume that the variable is restricted to those numbers for which the expression is defined. a. $x ^ { 7 }$
b. $\frac { 1 } { x ^ { 8 } }$
c. $x ^ { 8 }$
d. $\frac { 1 } { x ^ { 7 } }$

Answer 3

A

Question 4

Perform division and write the answer without using negative exponents. $\frac { - 12 x ^ { 6 } y ^ { 4 } z ^ { 9 } } { 3 x ^ { 9 } y ^ { 6 } z ^ { 0 } }$
a. $\frac { 4 z ^ { 9 } } { x ^ { 3 } y ^ { 6 } }$
b. $\frac { - 4 z ^ { 4 } } { x ^ { 3 } y ^ { 2 } }$
c. $\frac { 4 z ^ { 9 } } { x ^ { 3 } y ^ { 2 } }$
d. $\frac { - 4 z ^ { 9 } } { x ^ { 3 } y ^ { 2 } }$

Question 5

We can often multiply and divide radicals with different indexes.For example: $$\sqrt { 3 } \sqrt [ 3 ] { 5 } = \sqrt [ 6 ] { 27 } \sqrt [ 6 ] { 25 } = \sqrt [ 6 ] { ( 27 ) ( 25 ) } = \sqrt [ 6 ] { 675 }$$ Use this idea to write the following expression as a single radical. $\frac { \sqrt [ 4 ] { 2 } } { \sqrt { 6 } }$
a. $\frac { \sqrt [ 4 ] { 72 } } { 6 }$
b. $\frac { \sqrt [ 5 ] { 72 } } { 6 }$
c. $\frac { \sqrt [ 4 ] { 72 } } { 8 }$
d. $\frac { \sqrt [ 4 ] { 72 } } { 2 }$

Question 6

Simplify the expression. $$\left( x ^ { 5 } \right) ^ { 4 } \left( x ^ { 3 } \right) ^ { 3 }$$
a. $x ^ { 15 }$
b. $x ^ { 29 }$
c. $\quad x ^ { 6 }$

Question 7

Perform the operations and simplify. $$\frac { 2 a } { 13 } \cdot \frac { 3 } { 5 b }$$ Assume that no denominators are 0. a. $\frac { 3 } { 2 }$
b. $\frac { 13 a } { b 5 }$
c. $\frac { 6 a } { 65 b }$
d. $\frac { 2 } { 3 }$

Question 8

Simplify the radical expression. $\sqrt [ 6 ] { 8 }$
a. $\sqrt [ 18 ] { 2 }$
b. $\sqrt [ 2 ] { 200 }$
c. $\quad \sqrt [ 6 ] { 2 }$
d. $\quad \sqrt [ 2 ] { 2 }$
e. $\sqrt [ 2 ] { 8 }$

Question 9

Simplify the expression. $$\frac { \left( 8 ^ { - 2 } z ^ { - 4 } y \right) ^ { - 1 } } { \left( 5 y ^ { 3 } z ^ { - 3 } \right) ^ { 4 } \left( 5 y z ^ { - 3 } \right) ^ { - 1 } }$$ Write the answer without using negative exponents.Assume that all variables are restricted to those numbers for which the expression is defined. a. $\frac{8 z^{12}}{125 y^{13}}$
b. $\frac{64 z^{12}}{125 y^{13}}$
c. $\frac{125 y^{12}}{64 z^{13}}$
d. $\quad \frac { 64 z ^ { 13 } } { 125 y ^ { 12 } }$

Question 10

Select the correct representation of the inequality in interval notation. $x \leq 9$
a. $[ 9 , \infty )$
b. $[ - \infty , 9 ]$
c. $\quad ( 9 , \infty )$
d. $\quad ( - \infty , 9 ]$
e. $\quad ( - \infty , 9 )$

Question 11

Simplify the expression. $- 16 ^ { 3 / 2 }$
a. $- 192$
b. 67
c. $- 64$
d. $- 66$
e. $-128$
f. $- 24$

Question 12

Perform the division and write the answer without using negative exponents. $$\frac { 160 x ^ { 5 } y ^ { 7 } - 96 x ^ { 2 } y ^ { 5 } + 32 x y } { 4 x ^ { 5 } y ^ { 4 } }$$
a. $24 y ^ { 3 } - \frac { 40 y } { x ^ { 3 } } + \frac { 32 } { x ^ { 4 } y ^ { 3 } }$
b. $\quad 40 y ^ { 3 } - \frac { 40 y } { x ^ { 4 } } + \frac { 32 } { x ^ { 4 } y ^ { 3 } }$
c. $\quad 24 y ^ { 3 } - \frac { 24 y } { x ^ { 3 } } + \frac { 32 } { x ^ { 4 } y ^ { 9 } }$
d. $\quad 40 y ^ { 3 } - \frac { 24 y } { x ^ { 3 } } + \frac { 8 } { x ^ { 4 } y ^ { 3 } }$

Question 13

Simplify the expression. $( - 14 x ) ^ { 0 }$
Write the answer without using exponents.
a. $- 14$
b. $- 1$
c. 1
d. 14

Question 14

Multiply the expression as you would multiply polynomials. $$\left( x ^ { 17 / 2 } + y ^ { 7 / 2 } \right) ^ { 2 }$$
a. $\quad x ^ { 17 } - 2 x ^ { 17 } y ^ { 7 } + y ^ { 7 }$
b. $\quad x ^ { 17 } + x ^ { 17 } y ^ { 7 } + y ^ { 7 }$
c. $\quad x ^ { 17 } + y ^ { 7 }$
d. $\quad x ^ { 17 } + 2 x ^ { 17 / 2 } y ^ { 7 / 2 } + y ^ { 7 }$

Question 15

Simplify the expression.Assume that all variables represent positive numbers, so that no absolute value symbols are needed. $$\sqrt [ 4 ] { 2 x y ^ { 5 } } + y \sqrt [ 4 ] { 512 x y } - \sqrt [ 4 ] { 2 x y ^ { 5 } }$$
a. $\quad 8 y \sqrt { 3 x y }$
b. $\quad 12 y \sqrt { 2 x y }$
c. $\quad 4 y \sqrt { 4 x y }$
d. $\quad 4 y \sqrt { 2 x y }$

Question 16

Rationalize the denominator and simplify. $\frac { 2 } { \sqrt [ 3 ] { 2 } }$
a. $\sqrt [ 3 ] { 104 }$
b. $\sqrt [ 3 ] { 7 }$
c. $\sqrt [ 4 ] { 5 }$
d. $\quad \sqrt [ 3 ] { 4 }$
e. $\sqrt [ 6 ] { 4 }$

Question 17

Give the degree of the polynomial. $\sqrt { 791 }$
a. $1 / 2$
b. 0
c. This is not a polynomial
d. No defined degree

Question 18

Rationalize the numerator and simplify. $\frac { \sqrt { 5 } } { 20 }$
a. $\quad \frac { 1 } { 4 \sqrt { 3 } }$
b. $\quad \frac { 1 } { 4 \sqrt { 5 } }$
c. $\quad \frac { 1 } { 8 \sqrt { 3 } }$
d. $\frac { 1 } { 4 \sqrt { 9 } }$
e. $\frac { 1 } { 5 \sqrt { 5 } }$

Question 19

Simplify the expression. $$\left( \frac { 7 x ^ { - 5 } y ^ { 3 } z ^ { - 4 } } { 28 x ^ { 6 } y ^ { 11 } z ^ { - 9 } } \right) ^ { 3 }$$ Write the answer without using negative exponents. Assume that all variables are restricted to those numbers for which the expression is defined. a. $\frac { z ^ { 5 } } { 4 x ^ { 11 } y ^ { 8 } }$
b.
$\frac { z ^ { 15 } } { 64 x ^ { 24 } y ^ { 33 } }$
c. $\frac { z ^ { 15 } } { 64 x ^ { 33 } y ^ { 24 } }$
d. $\frac { z ^ { 15 } } { 64 x ^ { - 33 } y ^ { - 24 } }$
e. $\frac { z ^ { 5 } } { 4 x ^ { 33 } y ^ { 24 } }$

Question 20

Perform the operation and simplify. $- 3 a ^ { 2 } ( a + 1 ) + 9 a \left( a ^ { 2 } - 6 \right) - a ^ { 2 } ( a + 6 )$
a. $\quad 5 a ^ { 3 } - 9 a ^ { 2 } - 54 a$
b. $\quad 5 a ^ { 3 } - 9 a ^ { 2 } - 54$
c. $\quad 5 a ^ { 2 } + 9 a ^ { 4 } - 54$
d. 0

Question 21

Simplify the expression. $$\frac { 1 } { x ^ { - 7 } }$$
Write the answer without using negative exponents. Assume that the variable is restricted to those numbers for which the expression is defined.
a. $x ^ { 7 }$
b. $\quad \frac { 1 } { x ^ { 7 } }$
c. $x ^ { 8 }$
d. $\frac { 1 } { x ^ { 8 } }$

Question 22

Simplify the expression. $$\frac { x ^ { 6 } x ^ { 4 } } { x ^ { 3 } x }$$
Write the answer without using negative exponents. Assume that the variable is restricted to those numbers for which the expression is defined.
$\begin{array} { l l } \text { a. } & x ^ { 8 } \\ \text { b. } & x ^ { 7 } \\ \text { c. } & x ^ { 14 } \\ \text { d. } & x ^ { 6 } \end{array}$

Question 23

How many prime numbers are there between -6 and 14 on the number line? a. 19
b. 13
c. $\quad 0$
d. 14
e. 6
f. 5

Question 24

Simplify the expression. $$\left( x ^ { 3 } \right) ^ { 3 } \left( x ^ { 3 } \right) ^ { 2 }$$
a. $x ^ { 11 }$
b. $x ^ { - 1 }$
c. $x ^ { 10 }$
d. $x ^ { 15 }$

Question 25

Simplify the expression. $$\left( \frac { 8 x ^ { - 4 } y ^ { 5 } z ^ { - 8 } } { 32 x ^ { 4 } y ^ { 12 } z ^ { - 13 } } \right) ^ { 3 }$$
Write the answer without using negative exponents. Assume that all variables are restricted to those numbers for which the expression is defined.
a. $\quad \frac { z ^ { 5 } } { 4 x ^ { 8 } y ^ { 7 } }$
b. $\quad \frac { z ^ { 15 } } { 64 x ^ { 24 } y ^ { 21 } }$
c. $\quad \frac { z ^ { 15 } } { 64 x ^ { 21 } y ^ { 24 } }$
d. $\frac { z ^ { 15 } } { 64 x ^ { - 24 } y ^ { - 21 } }$
e. $\frac { z ^ { 5 } } { 4 x ^ { 24 } y ^ { 21 } }$

Question 26

Simplify the expression. $- 25 ^ { 4 / 2 }$
a. $- 1,250$
b. 628
c. $- 1,875$
d. $- 625$
e. $\quad - 50$
f. $- 627$

Question 27

Simplify the complex fraction. $$\frac { \frac { 4 x ^ { 2 } } { y ^ { 4 } } } { \frac { 8 x ^ { 3 } z ^ { 3 } } { y ^ { 2 } } }$$ Assume that the denominators are not 0. a. $\quad \frac { 1 } { 2 } x ^ { - 1 } y ^ { - 2 } z ^ { - 3 }$
b. $\quad \frac { 1 } { 2 } x ^ { 2 } y ^ { 3 } z ^ { 3 }$
c. $\quad \frac { 1 } { 2 } x ^ { 2 } y ^ { - 2 } z ^ { - 3 }$
d. $\frac { 1 } { 2 } x ^ { - 1 } y ^ { 4 } z ^ { - 3 }$

Question 28

We can often multiply and divide radicals with different indexes.For example: $$\sqrt { 3 } \sqrt [ 3 ] { 5 } = \sqrt [ 6 ] { 27 } \sqrt [ 6 ] { 25 } = \sqrt [ 6 ] { ( 27 ) ( 25 ) } = \sqrt [ 6 ] { 675 }$$
Use this idea to write the following expression as a single radical.
$\frac { \sqrt [ 6 ] { 4 } } { \sqrt { 5 } }$
a. $\quad \frac { \sqrt [ 6 ] { 500 } } { 9 }$
b. $\frac { \sqrt [ 6 ] { 500 } } { 5 }$
c. $\frac { \sqrt [ 6 ] { 500 } } { 4 }$
d. $\frac { \sqrt [ 7 ] { 500 } } { 5 }$

Question 29

Simplify the expression. $$\frac { \left( 8 ^ { - 2 } z ^ { - 5 } y \right) ^ { - 1 } } { \left( 5 y ^ { 5 } z ^ { - 1 } \right) ^ { 3 } \left( 5 y z ^ { - 1 } \right) ^ { - 2 } }$$
Write the answer without using negative exponents. Assume that all variables are restricted to those numbers for which the expression is defined.
a. $\quad \frac { 64 z ^ { 6 } } { 5 y ^ { 14 } }$
b. $\frac { 5 y ^ { 14 } } { 64 z ^ { 6 } }$
c. $\quad \frac { 64 z ^ { 14 } } { 5 y ^ { 6 } }$
d. $\frac { 8 z ^ { 5 } } { 5 y ^ { 15 } }$

Question 30

Simplify each complex fraction. $$\frac { x + 1 - \frac { 6 } { x } } { x + 5 + \frac { 6 } { x } }$$
Assume that no denominators are 0 .
a. $\frac { x + 3 } { x - 3 }$
b. $\frac { x - 2 } { x + 2 }$
c. $\frac { x + 2 } { x - 2 }$
d. $\frac { x - 3 } { x + 3 }$

Question 31

Simplify the fraction. $$\frac { x y + 6 x + 9 y + 54 } { x ^ { 3 } + 729 }$$ Assume that denominator is not 0 .
a. $\frac { y - 6 } { x ^ { 2 } - 9 x - 81 }$
b. $\frac { y + 9 } { x ^ { 2 } - 9 x + 81 }$
c. $\quad \frac { y + 6 } { x ^ { 2 } - 9 x + 81 }$
d. $\frac { y - 6 } { x ^ { 2 } - 9 x + 81 }$

Question 32

Simplify the expression.Assume that all variables represent positive numbers, so that no absolute value symbols are needed. $$\sqrt [ 4 ] { 2 x y ^ { 5 } } + y \sqrt [ 4 ] { 512 x y } - \sqrt [ 4 ] { 2 x y ^ { 5 } }$$
a. $\quad 4 y \sqrt { 4 x y }$
b. $\quad 4 y \sqrt { 2 x y }$
c. $\quad 12 y \sqrt { 2 x y }$
d. $\quad 8 y \sqrt { 3 x y }$

Question 33

Rationalize the numerator and simplify. $\frac { \sqrt { 5 } } { 25 }$
a. $\quad \frac { 1 } { 10 \sqrt { 2 } }$
b. $\quad \frac { 1 } { 5 \sqrt { 5 } }$
c. $\quad \frac { 1 } { 5 \sqrt { 2 } }$
d. $\quad \frac { 1 } { 6 \sqrt { 5 } }$
e. $\frac { 1 } { 5 \sqrt { 10 } }$

Question 34

Select the correct representation of the inequality in interval notation. $x \leq 3$
a. $\quad ( - \infty , 3 )$
b. $\quad ( - \infty , 3 ]$
c. $[ 3 , \infty )$
d. $\quad [ - \infty , 3 ]$
e. $\quad ( 3 , \infty )$

Question 35

Perform division and write the answer without using negative exponents. $\frac { - 190 x ^ { 6 } y ^ { 4 } z ^ { 9 } } { 19 x ^ { 9 } y ^ { 6 } z ^ { 0 } }$
a. $\frac { 10 z ^ { 9 } } { x ^ { 3 } y ^ { 6 } }$
b. $\quad \frac { 10 z ^ { 9 } } { x ^ { 3 } y ^ { 2 } }$
c. $\frac { - 10 z ^ { 4 } } { x ^ { 3 } y ^ { 2 } }$
d. $\frac { - 10 z ^ { 9 } } { x ^ { 3 } y ^ { 2 } }$

Question 36

Simplify the expression. $( - 13 x ) ^ { 0 }$
Write the answer without using exponents.
a. 1
b. 13
c. $- 13$
d. $- 1$

Question 37

Simplify the radical expression. $\sqrt [ 4 ] { 4 }$
a. $\sqrt [ 2 ] { 4 }$
b. $\sqrt [ 2 ] { 2 }$
c. $\sqrt [ 2 ] { 200 }$
d. $\sqrt [ 4 ] { 2 }$
e. $\sqrt [ 8 ] { 2 }$

Question 38

Perform the division and write the answer without using negative exponents. $$\frac { 100 x ^ { 5 } y ^ { 7 } - 60 x ^ { 2 } y ^ { 5 } + 20 x y } { 10 x ^ { 5 } y ^ { 4 } }$$
a. $\quad 6 y ^ { 3 } - \frac { 6 y } { x ^ { 3 } } + \frac { 20 } { x ^ { 4 } y ^ { 9 } }$
b. $\quad 10 y ^ { 3 } - \frac { 10 y } { x ^ { 4 } } + \frac { 20 } { x ^ { 4 } y ^ { 3 } }$
c. $10 y ^ { 3 } - \frac { 6 y } { x ^ { 3 } } + \frac { 2 } { x ^ { 4 } y ^ { 3 } }$
d. $6 y ^ { 3 } - \frac { 10 y } { x ^ { 3 } } + \frac { 20 } { x ^ { 4 } y ^ { 3 } }$

Question 39

Rationalize the denominator and simplify. $\frac { 4 } { \sqrt [ 4 ] { 4 } }$
a. $\sqrt [ 4 ] { 164 }$
b. $\sqrt [ 5 ] { 65 }$
c. $\sqrt [ 4 ] { 67 }$
d. $\quad \sqrt [ 4 ] { 64 }$
e. $\sqrt [ 8 ] { 64 }$

Question 40

Perform the operations and simplify. $$\frac { 1 } { x - 4 } + \frac { 3 } { x + 4 } - \frac { 3 x - 4 } { x ^ { 2 } - 16 }$$
Assume that no denominators are 0 .
a. $\frac { 4 } { x + 4 }$
b. $\frac { 1 } { x + 16 }$
c. $\frac { 1 } { x + 4 }$
d. $\frac { 1 } { x - 4 }$

Question 41

Simplify the expression. $$\left( \frac { 5 x ^ { - 5 } y ^ { 5 } z ^ { - 7 } } { 20 x ^ { 6 } y ^ { 10 } z ^ { - 12 } } \right) ^ { 3 }$$ Write the answer without using negative exponents. Assume that all variables are restricted to those numbers for which the expression is defined. a. $\frac { z ^ { 5 } } { 4 x ^ { 33 } y ^ { 15 } }$
b.
$\frac { z ^ { 15 } } { 64 x ^ { 33 } y ^ { 15 } }$
c. $\quad \frac { z ^ { 15 } } { 64 x ^ { 15 } y ^ { 33 } }$
d. $\frac { z ^ { 5 } } { 4 x ^ { 11 } y ^ { 5 } }$
e. $\frac { z ^ { 15 } } { 64 x ^ { - 33 } y ^ { - 15 } }$

Question 42

Perform the operations and simplify. $$\frac { 1 } { x - 4 } + \frac { 3 } { x + 4 } - \frac { 3 x - 4 } { x ^ { 2 } - 16 }$$
Assume that no denominators are 0 .
a. $\frac { 1 } { x + 16 }$
b. $\frac { 1 } { x + 4 }$
c. $\frac { 1 } { x - 4 }$
d. $\frac { 4 } { x + 4 }$

Question 43

Simplify the complex fraction. $\frac { \frac { 3 x ^ { 5 } } { y ^ { 2 } } } { \frac { 6 x ^ { 2 } z ^ { 4 } } { y ^ { 4 } } }$
Assume that the denominators are not 0 .
a. $\quad \frac { 1 } { 2 } x ^ { 3 } y ^ { 2 } z ^ { - 4 }$
b. $\frac { 1 } { 2 } x ^ { 4 } y ^ { 2 } z ^ { 4 }$
c. $\quad \frac { 1 } { 2 } x ^ { 5 } y ^ { 2 } z ^ { - 4 }$
d. $\quad \frac { 1 } { 2 } x ^ { 3 } y ^ { 2 } z ^ { - 4 }$

Question 44

Factor the expression completely. $$36 z ^ { 2 } + 84 z + 49$$
a. $\quad ( 6 z + 7 ) ^ { 2 }$
b. $\quad ( 6 z - 7 ) ^ { 2 }$
c. $\quad ( 6 z + 7 ) ( 6 z - 7 )$
d. $\quad 7 ( 6 z + 7 )$

Question 45

Perform the operations and simplify. $$\frac { 23 a } { 2 } \cdot \frac { 11 } { 2 b }$$
Assume that no denominators are 0 .
a. $\quad \frac { 11 } { 23 }$
b. $\frac { 253 a } { 4 b }$
c. $\frac { 23 } { 11 }$
d. $\frac { 2 a } { b 2 }$

Question 46

Simplify the expression. $$\frac { \left( 8 ^ { - 2 } z ^ { - 5 } y \right) ^ { - 2 } } { \left( 5 y ^ { 2 } z ^ { - 5 } \right) ^ { 5 } \left( 5 y z ^ { - 5 } \right) ^ { - 1 } }$$
Write the answer without using negative exponents. Assume that all variables are restricted to those numbers for which the expression is defined.
a. $\quad \frac { 512 z ^ { 29 } } { 625 y ^ { 12 } }$
b. $\quad \frac { 625 y ^ { 11 } } { 4096 z ^ { 30 } }$
c. $\quad \frac { 4096 z ^ { 30 } } { 625 y ^ { 11 } }$
d. $\frac { 4096 z ^ { 11 } } { 625 y ^ { 30 } }$

Question 47

Simplify the expression. $$\frac { 1 } { x ^ { - 8 } }$$
Write the answer without using negative exponents. Assume that the variable is restricted to those numbers for which the expression is defined.
a. $\frac { 1 } { x ^ { 9 } }$
b. $x ^ { 8 }$
c. $x ^ { 9 }$
d. $\frac { 1 } { x ^ { 8 } }$

Question 48

Perform the operation and simplify. $- 3 a ^ { 2 } ( a + 1 ) + 6 a \left( a ^ { 2 } - 4 \right) - a ^ { 2 } ( a + 10 )$
a. 0
b. $\quad 2 a ^ { 3 } - 13 a ^ { 2 } - 24$
c. $\quad 2 a ^ { 3 } - 13 a ^ { 2 } - 24 a$
d. $\quad 2 a ^ { 2 } + 13 a ^ { 4 } - 24$

Question 49

Simplify each complex fraction. $$\frac { x + 1 - \frac { 6 } { x } } { x + 5 + \frac { 6 } { x } }$$
Assume that no denominators are 0 .
a. $\frac { x - 2 } { x + 2 }$
b. $\frac { x + 2 } { x - 2 }$
c. $\frac { x - 3 } { x + 3 }$
d. $\frac { x + 3 } { x - 3 }$

Question 50

Simplify the radical expression. $\sqrt [ 4 ] { 49 }$
a. $\sqrt [ 8 ] { 7 }$
b. $\sqrt [ 2 ] { 700 }$
c. $\sqrt [ 2 ] { 49 }$
d. $\sqrt [ 4 ] { 7 }$
e. $\sqrt [ 2 ] { 7 }$

Question 51

Give the degree of the polynomial. $\sqrt { 576 }$
a. 0
b. No defined degree
c. $1 / 2$
d. This is not a polynomial

Question 52

Simplify the fraction. $\frac { x y + 6 x + 4 y + 24 } { x ^ { 3 } + 64 }$
Assume that denominator is not 0 .
a. $\frac { y - 6 } { x ^ { 2 } - 4 x - 16 }$
b. $\frac { y + 6 } { x ^ { 2 } - 4 x + 16 }$
c. $\frac { y - 6 } { x ^ { 2 } - 4 x + 16 }$
d. $\frac { y + 4 } { x ^ { 2 } - 4 x + 16 }$

Question 53

Simplify the expression. $$\frac { x ^ { 7 } x ^ { 4 } } { x ^ { 4 } x }$$
Write the answer without using negative exponents. Assume that the variable is restricted to those numbers for which the expression is defined.
$\begin{array} { l l } \text { a. } & x ^ { 16 } \\ \text { b. } & x ^ { 6 } \\ \text { c. } & x ^ { 7 } \\ \text { d. } & x ^ { 8 } \end{array}$

Question 54

How many prime numbers are there between -5 and 20 on the number line? a. 0
b. 4
c. 19
d. 24
e. 20
f. 8

Question 55

We can often multiply and divide radicals with different indexes.For example: $$\sqrt { 3 } \sqrt [ 3 ] { 5 } = \sqrt [ 6 ] { 27 } \sqrt [ 6 ] { 25 } = \sqrt [ 6 ] { ( 27 ) ( 25 ) } = \sqrt [ 6 ] { 675 }$$ Use this idea to write the following expression as a single radical. $\frac { \sqrt [ 6 ] { 4 } } { \sqrt { 7 } }$
a.
b. $\frac { \sqrt [ 7 ] { 1372 } } { 7 }$
c. $\frac { \sqrt [ 6 ] { 1372 } } { 4 }$
d. $\quad \frac { \sqrt [ 6 ] { 1372 } } { 7 }$

Question 56

Simplify the expression. $( - 4 x ) ^ { 0 }$
Write the answer without using exponents.
a. 1
b. 4
$\begin{array} { l l } \text { c. } & - 4 \\ \text { d. } & - 1 \end{array}$

Question 57

Rationalize the numerator and simplify. $\frac { \sqrt { 2 } } { 8 }$
a. $\quad \frac { 1 } { 8 \sqrt { 5 } }$
b. $\frac { 1 } { 4 \sqrt { 2 } }$
c. $\frac { 1 } { 4 \sqrt { 6 } }$
d. $\quad \frac { 1 } { 5 \sqrt { 2 } }$
e. $\frac { 1 } { 4 \sqrt { 5 } }$

Question 58

Multiply the expression as you would multiply polynomials. $$\left( x ^ { 7 / 2 } + y ^ { 9 / 2 } \right) ^ { 2 }$$
a. $\quad x ^ { 7 } + 2 x ^ { 7 / 2 } y ^ { 9 / 2 } + y ^ { 9 }$
b. $\quad x ^ { 7 } - 2 x ^ { 7 } y ^ { 9 } + y ^ { 9 }$
c. $\quad x ^ { 7 } + x ^ { 7 } y ^ { 9 } + y ^ { 9 }$
d. $\quad x ^ { 7 } + y ^ { 9 }$

Question 59

Simplify the expression. $$\left( x ^ { 6 } \right) ^ { 4 } \left( x ^ { 2 } \right) ^ { 2 }$$
a. $x ^ { - 8 }$
b. $x ^ { 6 }$
c. $x ^ { 14 }$
d. $x ^ { 28 }$

Question 60

Select the correct representation of the inequality in interval notation. $x \leq 7$
a. $\quad [ - \infty , 7 ]$
b. $\quad ( 7 , \infty )$
c. $[ 7 , \infty )$
d. $\quad ( - \infty , 7 )$
e. $\quad ( - \infty , 7 ]$

Question 61

Simplify the expression. $- 8 ^ { 4 / 3 }$
a. $- 10.6667$
b. 19
c. $- 32$
d. $- 48$
e. $- 18$
f. $- 16$

Question 62

Perform the operations and simplify. $$\frac { 1 } { x - 6 } + \frac { 3 } { x + 6 } - \frac { 3 x - 6 } { x ^ { 2 } - 36 }$$
Assume that no denominators are 0 .
a. $\frac { 1 } { x - 6 }$
b. $\frac { 1 } { x + 36 }$
c. $\frac { 1 } { x + 6 }$
d. $\frac { 6 } { x + 6 }$

Question 63

Give the degree of the polynomial. $\sqrt { 127 }$
a. $1 / 2$
b. No defined degree
c. This is not a polynomial
d. 0

Question 64

Identify the correct union of intervals for the inequality. $x \leq - 16$ or $x > 5$
a. $\quad ( - \infty , - 16 ] \cup ( 5 , \infty )$
b. $\quad ( - \infty , - 16 ) \cup [ 5 , \infty )$
c. $\quad ( - \infty , - 16 ) \cup ( 5 , \infty )$
d. $\quad ( - \infty , - 16 ] \cup ( 5 , \infty ]$
e. $\quad ( - \infty , - 16 ] \cup [ 5 , \infty )$

Question 65

How many natural numbers are there between -16.5 and 6.5 on the number line? a. $\quad 0$
b. 7
c. 12
d. 6
e. 23

Question 66

Simplify each complex fraction. $$\frac { x + 2 - \frac { 63 } { x } } { x + 16 + \frac { 63 } { x } }$$ Assume that no denominators are 0 .
a. $\frac { x + 9 } { x - 9 }$
b. $\frac { x - 7 } { x + 7 }$
c. $\frac { x + 7 } { x - 7 }$
d. $\frac { x - 9 } { x + 9 }$

Question 67

Perform the operations and simplify. $$\frac { 5 a } { 3 } \cdot \frac { 2 } { 7 b }$$
Assume that no denominators are 0 .
a. $\frac { 3 a } { b 7 }$
b. $\frac { 5 } { 2 }$
c. $\frac { 2 } { 5 }$
d. $\frac { 10 a } { 21 b }$

Question 68

Rationalize the denominator and simplify. $\frac { 3 } { \sqrt [ 5 ] { 3 } }$
a. $\sqrt [ 6 ] { 82 }$
b. $\sqrt [ 5 ] { 181 }$
c. $\quad \sqrt [ 5 ] { 84 }$
d. $\sqrt [ 10 ] { 81 }$
e. $\sqrt [ 5 ] { 81 }$

Question 69

Write the expression without using absolute value symbols. $| x + 4 | - | x - 11 | \quad$ for $\quad x < - 8$
$| x + 4 | - | x - 11 | =ــــــــــــ$ for $x < - 8$
a. 15
b. $2 x - 15$
c. 7
d. $15 - 2 x$
e. $- 15$

Question 70

Perform division and write the answer without using negative exponents. $\frac { - 56 x ^ { 6 } y ^ { 4 } z ^ { 9 } } { 14 x ^ { 9 } y ^ { 6 } z ^ { 0 } }$
a. $\frac { - 4 z ^ { 4 } } { x ^ { 3 } y ^ { 2 } }$
b. $\frac { - 4 z ^ { 9 } } { x ^ { 3 } y ^ { 2 } }$
c. $\frac { 4 z ^ { 9 } } { x ^ { 3 } y ^ { 6 } }$
d. $\frac { 4 z ^ { 9 } } { x ^ { 3 } y ^ { 2 } }$

Question 71

Simplify the complex fraction. $$\frac { \frac { 2 x ^ { 2 } } { y ^ { 3 } } } { \frac { 4 x ^ { 4 } z ^ { 3 } } { y ^ { 5 } } }$$ Assume that the denominators are not 0. a. $\quad \frac { 1 } { 2 } x ^ { 2 } y ^ { 2 } z ^ { - 3 }$
b. $\quad \frac { 1 } { 2 } x ^ { - 2 } y ^ { 3 } z ^ { - 3 }$
c. $\quad \frac { 1 } { 2 } x ^ { - 2 } y ^ { 2 } z ^ { - 3 }$
d. $\frac { 1 } { 2 } x ^ { 5 } y ^ { 4 } z ^ { 3 }$

Question 72

Simplify the expression.Assume that all variables represent positive numbers, so that no absolute value symbols are needed. $$\sqrt [ 4 ] { 2 x y ^ { 5 } } + y \sqrt [ 4 ] { 512 x y } - \sqrt [ 4 ] { 2 x y ^ { 5 } }$$
a. $\quad 12 y \sqrt { 2 x y }$
b. $\quad 8 y \sqrt { 3 x y }$
c. $\quad 4 y \sqrt { 2 x y }$
d. $\quad 4 y \sqrt { 4 x y }$

Question 73

Perform the operation and simplify. $- 3 a ^ { 2 } ( a + 1 ) + 7 a \left( a ^ { 2 } - 4 \right) - a ^ { 2 } ( a + 9 )$
a. $\quad 3 a ^ { 2 } + 12 a ^ { 4 } - 28$
b. $\quad 3 a ^ { 3 } - 12 a ^ { 2 } - 28$
c. 0
d. $\quad 3 a ^ { 3 } - 12 a ^ { 2 } - 28 a$

Question 74

Factor the expression completely. $9 z ^ { 2 } + 42 z + 49$
a. $\quad ( 3 z - 7 ) ^ { 2 }$
b. $\quad ( 3 z + 7 ) ( 3 z - 7 )$
c. $\quad 7 ( 3 z + 7 )$
d. $\quad ( 3 z + 7 ) ^ { 2 }$

Question 75

Simplify the fraction. $\frac { x y + 6 x + 8 y + 48 } { x ^ { 3 } + 512 }$
Assume that denominator is not 0 .
a. $\frac { y - 6 } { x ^ { 2 } - 8 x + 64 }$
b. $\frac { y + 8 } { x ^ { 2 } - 8 x + 64 }$
c. $\frac { y - 6 } { x ^ { 2 } - 8 x - 64 }$
d. $\frac { y + 6 } { x ^ { 2 } - 8 x + 64 }$

Question 76

Simplify the expression. $$\left( \frac { a ^ { - 5 } } { b ^ { - 3 } } \right) ^ { - 4 }$$
Write the answer without using negative exponents. Assume that all variables are restricted to those numbers for which the expression is defined.
a. $\quad \frac { a ^ { 12 } } { b ^ { 20 } }$
b. $\quad \frac { a ^ { 20 } } { b ^ { 12 } }$
c. $\quad \frac { b ^ { 12 } } { a ^ { 20 } }$
d. $\quad \frac { b ^ { 20 } } { a ^ { 12 } }$

Question 77

Simplify the expression. $$\left( \frac { r ^ { 5 } r ^ { - 1 } } { r ^ { 3 } r ^ { - 3 } } \right) ^ { 2 }$$
Write the answer without using negative exponents. Assume that the variable is restricted to those numbers for which the expression is defined.
a. $r ^ { 2 }$
b. $\quad r ^ { 8 }$
c. $\quad r ^ { 0 }$
d. $\quad r ^ { 12 }$

Question 78

Multiply the expression as you would multiply polynomials. $$\left( x ^ { 11 / 2 } + y ^ { 15 / 2 } \right) ^ { 2 }$$
a. $\quad x ^ { 11 } + 2 x ^ { 11 / 2 } y ^ { 15 / 2 } + y ^ { 15 }$
b. $\quad x ^ { 11 } + x ^ { 11 } y ^ { 15 } + y ^ { 15 }$
c. $\quad x ^ { 11 } - 2 x ^ { 11 } y ^ { 15 } + y ^ { 15 }$
d. $\quad x ^ { 11 } + y ^ { 15 }$

Question 79

Calculate the volume of a box that has dimensions of 6,000 by 8,600 by 4,800 millimeters. a. $\quad 2.4768 \times 10 ^ { 10 } \mathrm {~mm} ^ { 3 }$
b. $\quad 2.4768 \times 10 ^ { 11 } \mathrm {~mm} ^ { 3 }$
c. $\quad 1.9975 \times 10 ^ { 10 } \mathrm {~mm} ^ { 3 }$
d. $\quad 1.9975 \times 10 ^ { 11 } \mathrm {~mm} ^ { 3 }$

Question 80

Perform the division and write the answer without using negative exponents. $$\frac { 400 x ^ { 5 } y ^ { 7 } - 240 x ^ { 2 } y ^ { 5 } + 80 x y } { 16 x ^ { 5 } y ^ { 4 } }$$
a. $25 y ^ { 3 } - \frac { 15 y } { x ^ { 3 } } + \frac { 5 } { x ^ { 4 } y ^ { 3 } }$
b. $\quad 15 y ^ { 3 } - \frac { 15 y } { x ^ { 3 } } + \frac { 80 } { x ^ { 4 } y ^ { 9 } }$
c. $15 y ^ { 3 } - \frac { 25 y } { x ^ { 3 } } + \frac { 80 } { x ^ { 4 } y ^ { 3 } }$
d. $\quad 25 y ^ { 3 } - \frac { 25 y } { x ^ { 4 } } + \frac { 80 } { x ^ { 4 } y ^ { 3 } }$