Question 1

Simplify the expression. Assume all variables represent nonzero real numbers.
-$$\left( x ^ { 2 } y ^ { 4 } \right) ^ { 3 }$$

Accepted Answer

A)  $x^{8} y^{64}$ 
B)  $x ^ { 5 } y ^ { 7 }$ 
C)  $x^{6} y^{12}$ 
D)  $x ^ { 2 } y ^ { 12 }$ 
A)  $x^{8} y^{64}$ 
B)  $x ^ { 5 } y ^ { 7 }$ 
C)  $x^{6} y^{12}$ 
D)  $x ^ { 2 } y ^ { 12 }$

Question 2

Identify the expression as a polynomial or not a polynomial.
-9x - 12

Accepted Answer

A)  Not a polynomial 
B)  Polynomial 
A)  Not a polynomial 
B)  Polynomial

Question 3

Solve the problem.
-At an altitude of $h$ feet above the surface of the earth, the approximate distance in miles that a person can see is given by $\mathrm { d } = 1.2247 \sqrt { \mathrm { h } }$. How far can a person see if if he or she is 1260 feet above the earth's surface? Round your answer to the nearest tenth of a mile.

Accepted Answer

A)  $45.9 \mathrm { mi }$ 
B)  $49.2 \mathrm { mi }$ 
C)  $43.5 \mathrm { mi }$ 
D)  $44.6 \mathrm { mi }$ 
A)  $45.9 \mathrm { mi }$ 
B)  $49.2 \mathrm { mi }$ 
C)  $43.5 \mathrm { mi }$ 
D)  $44.6 \mathrm { mi }$

Question 4

Find the sum or difference.
-$$\left( 6 n ^ { 5 } + 9 n + 7 n ^ { 2 } \right) + \left( 3 n ^ { 2 } + 4 n ^ { 5 } + 3 n \right)$$

Accepted Answer

A)  $13 n ^ { 5 } + 9 n ^ { 2 } + 10 n$ 
B)  $10 n + 10 n ^ { 5 } + 12 n ^ { 2 }$ 
C)  $32 n ^ { 8 }$ 
D)  $10 n ^ { 5 } + 10 n ^ { 2 } + 12 n$ 
A)  $13 n ^ { 5 } + 9 n ^ { 2 } + 10 n$ 
B)  $10 n + 10 n ^ { 5 } + 12 n ^ { 2 }$ 
C)  $32 n ^ { 8 }$ 
D)  $10 n ^ { 5 } + 10 n ^ { 2 } + 12 n$

Question 5

Divide.
-$$\frac { x ^ { 2 } + 4 x - 21 } { x + 7 }$$

Accepted Answer

A)  $x + 3$ 
B)  $x ^ { 2 } - 3$ 
C)  $x ^ { 2 } + 5 x - 14$ 
D)  $x - 3$ 
A)  $x + 3$ 
B)  $x ^ { 2 } - 3$ 
C)  $x ^ { 2 } + 5 x - 14$ 
D)  $x - 3$

Question 6

Factor by grouping.
-$$8 x ^ { 6 } - 12 x ^ { 3 } + 10 x ^ { 3 } - 15$$

Accepted Answer

A)  $\left( 4 x ^ { 3 } - 5 \right) \left( 2 x ^ { 3 } + 3 \right)$ 
B)  $\left( 8 x ^ { 3 } - 5 \right) \left( x ^ { 3 } + 3 \right)$ 
C)  $\left( 4 x ^ { 6 } + 5 \right) ( 2 x - 3 )$ 
D)  $\left( 4 x ^ { 3 } + 5 \right) \left( 2 x ^ { 3 } - 3 \right)$ 
A)  $\left( 4 x ^ { 3 } - 5 \right) \left( 2 x ^ { 3 } + 3 \right)$ 
B)  $\left( 8 x ^ { 3 } - 5 \right) \left( x ^ { 3 } + 3 \right)$ 
C)  $\left( 4 x ^ { 6 } + 5 \right) ( 2 x - 3 )$ 
D)  $\left( 4 x ^ { 3 } + 5 \right) \left( 2 x ^ { 3 } - 3 \right)$

Question 7

Identify the set as finite or infinite.
-$$\left\{ 1 , \frac { 1 } { 3 } , \frac { 1 } { 9 } , \frac { 1 } { 27 } , \ldots , \right\}$$

Accepted Answer

A)  finite 
B)  infinite 
A)  finite 
B)  infinite

Question 8

Perform the indicated operations. Write the result using only positive exponents. Assume all variables represent nonzero
real numbers.
-$\frac { 16 r ^ { 5 } \left( r ^ { 2 } \right) ^ { - 5 } } { 9 \left( r ^ { 3 } \right) ^ { - 5 } }$

Accepted Answer

A)  $\frac { 16 } { 9 r ^ { 4 } }$ 
B)  $\frac { 16 r ^ { 4 } } { 9 }$ 
C)  $\frac { 16 r ^ { 10 } } { 9 }$ 
D)  $\frac { 16 } { 9 r ^ { 10 } }$ 
A)  $\frac { 16 } { 9 r ^ { 4 } }$ 
B)  $\frac { 16 r ^ { 4 } } { 9 }$ 
C)  $\frac { 16 r ^ { 10 } } { 9 }$ 
D)  $\frac { 16 } { 9 r ^ { 10 } }$

Question 9

Simplify the expression. Assume all variables represent positive real numbers.
-$\sqrt { 5 x } + 8 \sqrt { 80 x } + 5 \sqrt { 180 x }$

Accepted Answer

A)  $63 \sqrt { 5 x }$ 
B)  $13 \sqrt { 265 x }$ 
C)  $62 \sqrt { 5 x }$ 
D)  $14 \sqrt { 265 x }$ 
A)  $63 \sqrt { 5 x }$ 
B)  $13 \sqrt { 265 x }$ 
C)  $62 \sqrt { 5 x }$ 
D)  $14 \sqrt { 265 x }$

Question 10

Perform the indicated operation and write your answer with positive integer exponents.
-$$\frac { ( m n ) ^ { - 1 } } { m ^ { - 2 } + n ^ { - 2 } }$$

Accepted Answer

A)  $\frac { m ^ { 2 } + n ^ { 2 } } { m n }$ 
B)  $\frac { n } { n ^ { 2 } - m ^ { 2 } }$ 
C)  $\frac { \mathrm { mn } } { \mathrm { n } + \mathrm { m } }$ 
D)  $\frac { \mathrm { mn } } { \mathrm { n } ^ { 2 } + \mathrm { m } ^ { 2 } }$ 
A)  $\frac { m ^ { 2 } + n ^ { 2 } } { m n }$ 
B)  $\frac { n } { n ^ { 2 } - m ^ { 2 } }$ 
C)  $\frac { \mathrm { mn } } { \mathrm { n } + \mathrm { m } }$ 
D)  $\frac { \mathrm { mn } } { \mathrm { n } ^ { 2 } + \mathrm { m } ^ { 2 } }$

Question 11

Evaluate the expression.
-Let $x = - 8 , y = 4$. Evaluate $| x + y |$.

Accepted Answer

A)  4 
B)  $- 12$ 
C)  $- 4$ 
D)  12 
A)  4 
B)  $- 12$ 
C)  $- 4$ 
D)  12

Question 12

Rationalize the denominator. Assume that all variables represent nonnegative numbers and that the denominator is not
zero.
-$$\frac { 5 } { 8 - \sqrt { 2 } }$$

Accepted Answer

A)  $\frac { 40 + 5 \sqrt { 2 } } { 62 }$ 
B)  $\frac { 40 + 5 \sqrt { 2 } } { 6 }$ 
C)  $\frac { 40 - 5 \sqrt { 2 } } { 62 }$ 
D)  $\frac { 5 } { 8 } - \frac { 5 } { \sqrt { 2 } }$ 
A)  $\frac { 40 + 5 \sqrt { 2 } } { 62 }$ 
B)  $\frac { 40 + 5 \sqrt { 2 } } { 6 }$ 
C)  $\frac { 40 - 5 \sqrt { 2 } } { 62 }$ 
D)  $\frac { 5 } { 8 } - \frac { 5 } { \sqrt { 2 } }$

Question 13

Factor the trinomial, if possible.
-$$25 a ^ { 2 } + 50 a b + 25 b ^ { 2 }$$

Accepted Answer

A)  $25 ( a - b ) ^ { 2 }$ 
B)  $( 5 a - 5 b ) ^ { 2 }$ 
C)  $( 5 a + 5 b ) ^ { 2 }$ 
D)  $25 ( a + b ) ^ { 2 }$ 
A)  $25 ( a - b ) ^ { 2 }$ 
B)  $( 5 a - 5 b ) ^ { 2 }$ 
C)  $( 5 a + 5 b ) ^ { 2 }$ 
D)  $25 ( a + b ) ^ { 2 }$

Question 14

Find the sum or difference.
-$$- \left( 2 x ^ { 3 } + x - 7 \right) + \left( 4 x ^ { 3 } + 3 x ^ { 2 } \right) - \left( 7 x ^ { 2 } - 9 x - 1 \right)$$

Accepted Answer

A)  $6 x ^ { 3 } - 10 x ^ { 2 } + 8 x - 8$ 
B)  $1 x ^ { 3 } - 2 x ^ { 2 } - 8 x + 8$ 
C)  $2 x ^ { 3 } + 4 x ^ { 2 } + 10 x - 8$ 
D)  $2 x ^ { 3 } - 4 x ^ { 2 } + 8 x + 8$ 
A)  $6 x ^ { 3 } - 10 x ^ { 2 } + 8 x - 8$ 
B)  $1 x ^ { 3 } - 2 x ^ { 2 } - 8 x + 8$ 
C)  $2 x ^ { 3 } + 4 x ^ { 2 } + 10 x - 8$ 
D)  $2 x ^ { 3 } - 4 x ^ { 2 } + 8 x + 8$

Question 15

Factor by any method.
-$$20 z ^ { 6 } + 7 z ^ { 3 } - 6$$

Accepted Answer

A)  $\left( 4 z ^ { 3 } + 3 \right) \left( 5 z ^ { 3 } - 2 \right)$ 
B)  $20 \left( z ^ { 3 } - 2 \right) \left( z ^ { 3 } + 3 \right)$ 
C)  $\left( 4 z ^ { 3 } + 1 \right) \left( 5 z ^ { 3 } - 6 \right)$ 
D)  $\left( 5 z ^ { 3 } + 2 \right) \left( 4 z ^ { 3 } - 3 \right)$ 
A)  $\left( 4 z ^ { 3 } + 3 \right) \left( 5 z ^ { 3 } - 2 \right)$ 
B)  $20 \left( z ^ { 3 } - 2 \right) \left( z ^ { 3 } + 3 \right)$ 
C)  $\left( 4 z ^ { 3 } + 1 \right) \left( 5 z ^ { 3 } - 6 \right)$ 
D)  $\left( 5 z ^ { 3 } + 2 \right) \left( 4 z ^ { 3 } - 3 \right)$

Question 16

Simplify the expression. Assume all variables represent nonzero real numbers.
-$2 ^ { 6 } \cdot 2 ^ { 3 }$

Accepted Answer

A)  $2^{9}$ 
B) $4^{18}$ 
C) $2^{18}$ 
D) $4^{9}$ 
A)  $2^{9}$ 
B) $4^{18}$ 
C) $2^{18}$ 
D) $4^{9}$

Question 17

Write in radical form. Assume all variables represent positive real numbers.
-$6 x ^ { 1 / 3 }$

Accepted Answer

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

Question 18

Insert ⊆ or ⊈ in the blank to make the statement true.
-$\varnothing$ _____$\varnothing$

Accepted Answer

A)   
B)   
A)   
B)

Question 19

Factor the polynomial by substitution.
-$$8 x ^ { 4 } + 18 x ^ { 2 } + 9$$

Accepted Answer

A)  $\left( 4 x ^ { 2 } + 3 \right) \left( 2 x ^ { 2 } + 3 \right)$ 
B)  $\left( 8 x ^ { 2 } + 3 \right) \left( x ^ { 2 } + 3 \right)$ 
C)  $\left( 4 x ^ { 2 } - 3 \right) \left( 2 x ^ { 2 } - 3 \right)$ 
D)  $\left( 4 x ^ { 2 } + 1 \right) \left( 2 x ^ { 2 } + 9 \right)$ 
A)  $\left( 4 x ^ { 2 } + 3 \right) \left( 2 x ^ { 2 } + 3 \right)$ 
B)  $\left( 8 x ^ { 2 } + 3 \right) \left( x ^ { 2 } + 3 \right)$ 
C)  $\left( 4 x ^ { 2 } - 3 \right) \left( 2 x ^ { 2 } - 3 \right)$ 
D)  $\left( 4 x ^ { 2 } + 1 \right) \left( 2 x ^ { 2 } + 9 \right)$

Question 20

Find the product.
-$$( 4 a - 7 ) ^ { 2 }$$

Accepted Answer

A)  $16 a ^ { 2 } + 49$ 
B)  $4 a ^ { 2 } - 56 a + 49$ 
C)  $16 a ^ { 2 } - 56 a + 49$ 
D)  $4 a ^ { 2 } + 49$ 
A)  $16 a ^ { 2 } + 49$ 
B)  $4 a ^ { 2 } - 56 a + 49$ 
C)  $16 a ^ { 2 } - 56 a + 49$ 
D)  $4 a ^ { 2 } + 49$

Simplify the expression. Assume all variables represent nonzero real numbers. - $\left( x ^ { 2 } y ^ { 4 } \right) ^ { 3 }$

Identify the expression as a polynomial or not a polynomial. -9x - 12

Find the sum or difference. - $\left( 6 n ^ { 5 } + 9 n + 7 n ^ { 2 } \right) + \left( 3 n ^ { 2 } + 4 n ^ { 5 } + 3 n \right)$

Divide. - $\frac { x ^ { 2 } + 4 x - 21 } { x + 7 }$

Factor by grouping. - $8 x ^ { 6 } - 12 x ^ { 3 } + 10 x ^ { 3 } - 15$

Identify the set as finite or infinite. - $\left\{ 1 , \frac { 1 } { 3 } , \frac { 1 } { 9 } , \frac { 1 } { 27 } , \ldots , \right\}$

Perform the indicated operations. Write the result using only positive exponents. Assume all variables represent nonzero real numbers. - $\frac { 16 r ^ { 5 } \left( r ^ { 2 } \right) ^ { - 5 } } { 9 \left( r ^ { 3 } \right) ^ { - 5 } }$

Simplify the expression. Assume all variables represent positive real numbers. - $\sqrt { 5 x } + 8 \sqrt { 80 x } + 5 \sqrt { 180 x }$

Perform the indicated operation and write your answer with positive integer exponents. - $\frac { ( m n ) ^ { - 1 } } { m ^ { - 2 } + n ^ { - 2 } }$

Evaluate the expression. -Let $x = - 8 , y = 4$ . Evaluate $| x + y |$ .

Rationalize the denominator. Assume that all variables represent nonnegative numbers and that the denominator is not zero. - $\frac { 5 } { 8 - \sqrt { 2 } }$

Factor the trinomial, if possible. - $25 a ^ { 2 } + 50 a b + 25 b ^ { 2 }$

Find the sum or difference. - $- \left( 2 x ^ { 3 } + x - 7 \right) + \left( 4 x ^ { 3 } + 3 x ^ { 2 } \right) - \left( 7 x ^ { 2 } - 9 x - 1 \right)$

Factor by any method. - $20 z ^ { 6 } + 7 z ^ { 3 } - 6$

Simplify the expression. Assume all variables represent nonzero real numbers. - $2 ^ { 6 } \cdot 2 ^ { 3 }$

Write in radical form. Assume all variables represent positive real numbers. - $6 x ^ { 1 / 3 }$

Insert ⊆ or ⊈ in the blank to make the statement true. - $\varnothing$ _____ $\varnothing$

Factor the polynomial by substitution. - $8 x ^ { 4 } + 18 x ^ { 2 } + 9$

Find the product. - $( 4 a - 7 ) ^ { 2 }$

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Inverse, Exponential, and Logarithmic Functions

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Exam 12: Review of Basic Concepts

Simplify the expression. Assume all variables represent nonzero real numbers. - (x2y4)3\left( x ^ { 2 } y ^ { 4 } \right) ^ { 3 }(x2y4)3

Identify the expression as a polynomial or not a polynomial. -9x - 12

Find the sum or difference. - (6n5+9n+7n2)+(3n2+4n5+3n)\left( 6 n ^ { 5 } + 9 n + 7 n ^ { 2 } \right) + \left( 3 n ^ { 2 } + 4 n ^ { 5 } + 3 n \right)(6n5+9n+7n2)+(3n2+4n5+3n)

Divide. - x2+4x−21x+7\frac { x ^ { 2 } + 4 x - 21 } { x + 7 }x+7x2+4x−21​

Factor by grouping. - 8x6−12x3+10x3−158 x ^ { 6 } - 12 x ^ { 3 } + 10 x ^ { 3 } - 158x6−12x3+10x3−15

Identify the set as finite or infinite. - {1,13,19,127,…,}\left\{ 1 , \frac { 1 } { 3 } , \frac { 1 } { 9 } , \frac { 1 } { 27 } , \ldots , \right\}{1,31​,91​,271​,…,}

Perform the indicated operations. Write the result using only positive exponents. Assume all variables represent nonzero real numbers. - 16r5(r2)−59(r3)−5\frac { 16 r ^ { 5 } \left( r ^ { 2 } \right) ^ { - 5 } } { 9 \left( r ^ { 3 } \right) ^ { - 5 } }9(r3)−516r5(r2)−5​

Simplify the expression. Assume all variables represent positive real numbers. - 5x+880x+5180x\sqrt { 5 x } + 8 \sqrt { 80 x } + 5 \sqrt { 180 x }5x​+880x​+5180x​

Perform the indicated operation and write your answer with positive integer exponents. - (mn)−1m−2+n−2\frac { ( m n ) ^ { - 1 } } { m ^ { - 2 } + n ^ { - 2 } }m−2+n−2(mn)−1​

Evaluate the expression. -Let x=−8,y=4x = - 8 , y = 4x=−8,y=4 . Evaluate ∣x+y∣| x + y |∣x+y∣ .

Rationalize the denominator. Assume that all variables represent nonnegative numbers and that the denominator is not zero. - 58−2\frac { 5 } { 8 - \sqrt { 2 } }8−2​5​

Factor the trinomial, if possible. - 25a2+50ab+25b225 a ^ { 2 } + 50 a b + 25 b ^ { 2 }25a2+50ab+25b2

Find the sum or difference. - −(2x3+x−7)+(4x3+3x2)−(7x2−9x−1)- \left( 2 x ^ { 3 } + x - 7 \right) + \left( 4 x ^ { 3 } + 3 x ^ { 2 } \right) - \left( 7 x ^ { 2 } - 9 x - 1 \right)−(2x3+x−7)+(4x3+3x2)−(7x2−9x−1)

Factor by any method. - 20z6+7z3−620 z ^ { 6 } + 7 z ^ { 3 } - 620z6+7z3−6

Simplify the expression. Assume all variables represent nonzero real numbers. - 26⋅232 ^ { 6 } \cdot 2 ^ { 3 }26⋅23

Write in radical form. Assume all variables represent positive real numbers. - 6x1/36 x ^ { 1 / 3 }6x1/3

Insert ⊆ or ⊈ in the blank to make the statement true. - ∅\varnothing∅ _____ ∅\varnothing∅

Factor the polynomial by substitution. - 8x4+18x2+98 x ^ { 4 } + 18 x ^ { 2 } + 98x4+18x2+9

Find the product. - (4a−7)2( 4 a - 7 ) ^ { 2 }(4a−7)2