Question 1

Factor by grouping.
-$15 a ^ { 3 } + 9 a ^ { 2 } b + 25 a b ^ { 2 } + 15 b ^ { 3 }$

Accepted Answer

A)  $\left( 15 a ^ { 2 } + 5 b ^ { 2 } \right) ( a + 3 b )$ 
B)  $\left( 3 a ^ { 2 } - 5 b ^ { 2 } \right) ( 5 a - 3 b )$ 
C)  $\left( 3 a ^ { 2 } + 5 b \right) ( 5 a + 3 b )$ 
D)  $\left( 3 a ^ { 2 } + 5 b ^ { 2 } \right) ( 5 a + 3 b )$ 
A)  $\left( 15 a ^ { 2 } + 5 b ^ { 2 } \right) ( a + 3 b )$ 
B)  $\left( 3 a ^ { 2 } - 5 b ^ { 2 } \right) ( 5 a - 3 b )$ 
C)  $\left( 3 a ^ { 2 } + 5 b \right) ( 5 a + 3 b )$ 
D)  $\left( 3 a ^ { 2 } + 5 b ^ { 2 } \right) ( 5 a + 3 b )$

Question 2

Evaluate the expression.
--2[6 - 6(-9 - 6)]

Accepted Answer

A)  -4 
B) 78 
C) -192 
D)  0 
A)  -4 
B) 78 
C) -192 
D)  0

Question 3

Simplify the expression. Assume all variables represent positive real numbers.
-$\sqrt [ 3 ] { \frac { 8 } { 9 x ^ { 2 } } }$

Accepted Answer

A)  $\frac { \sqrt [ 3 ] { 648 x ^ { 2 } } } { 9 x }$ 
B)  $\frac { \sqrt [ 3 ] { 648 x } } { 81 }$ 
C)  $\frac { \sqrt [ 3 ] { 72 x } } { 9 x }$ 
D)  $\frac { \sqrt [ 3 ] { 24 x } } { 3 x }$ 
A)  $\frac { \sqrt [ 3 ] { 648 x ^ { 2 } } } { 9 x }$ 
B)  $\frac { \sqrt [ 3 ] { 648 x } } { 81 }$ 
C)  $\frac { \sqrt [ 3 ] { 72 x } } { 9 x }$ 
D)  $\frac { \sqrt [ 3 ] { 24 x } } { 3 x }$

Question 4

Simplify the expression. Assume all variables represent positive real numbers.
-$- \sqrt [ 4 ] { 625 }$

Accepted Answer

A)  $- 25$ 
B)  $- 5$ 
C)  $- 4$ 
D)  not a real number 
A)  $- 25$ 
B)  $- 5$ 
C)  $- 4$ 
D)  not a real number

Question 5

Factor the polynomial by substitution.
-$$9 y ^ { 4 } + 6 y ^ { 2 } - 8$$

Accepted Answer

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

Question 6

Identify the property illustrated by the statement. Assume all variables represent real numbers.
-9(x + 4) = 9x + 36

Accepted Answer

A)  Commutative 
B)  Distributive 
C)  Associative 
D)  Closure 
A)  Commutative 
B)  Distributive 
C)  Associative 
D)  Closure

Question 7

Perform the indicated operations. Write the result using only positive exponents. Assume all variables represent nonzero
real numbers.
-$\frac { x ^ { - 2 } } { x ^ { - 15 } }$

Accepted Answer

A)  $\frac { 1 } { x ^ { 17 } }$ 
B)  $\frac { 1 } { x ^ { 13 } }$ 
C)  $- x ^ { 13 }$ 
D)  $x ^ { 13 }$ 
A)  $\frac { 1 } { x ^ { 17 } }$ 
B)  $\frac { 1 } { x ^ { 13 } }$ 
C)  $- x ^ { 13 }$ 
D)  $x ^ { 13 }$

Question 8

Solve the problem.
-If the lengths of the sides of a square are tripled, by what factor will the area change?

Accepted Answer

A)  6 
B)  9 
C)  27 
D)  3 
A)  6 
B)  9 
C)  27 
D)  3

Question 9

Identify the property illustrated by the statement. Assume all variables represent real numbers.
-(9 + 1) + 1 = (1 + 9) + 1

Accepted Answer

A)  Closure 
B)  Commutative 
C)  Distributive 
D)  Associative 
A)  Closure 
B)  Commutative 
C)  Distributive 
D)  Associative

Question 10

Simplify the expression. Assume all variables represent positive real numbers.
-$- \sqrt { 147 }$

Accepted Answer

A)  $- 7 \sqrt { 3 }$ 
B)  12 
C)  not a real number 
D)  $- 49 \sqrt { 3 }$ 
A)  $- 7 \sqrt { 3 }$ 
B)  12 
C)  not a real number 
D)  $- 49 \sqrt { 3 }$

Question 11

Find the product.
-$$( 7 p - 1 ) \left( 49 p ^ { 2 } + 7 p + 1 \right)$$

Accepted Answer

A)  $49 p ^ { 3 } - 1$ 
B)  $343 p ^ { 3 } + 56 p ^ { 2 } - 1$ 
C)  $343 p ^ { 3 } + 1$ 
D)  $343 p ^ { 3 } - 1$ 
A)  $49 p ^ { 3 } - 1$ 
B)  $343 p ^ { 3 } + 56 p ^ { 2 } - 1$ 
C)  $343 p ^ { 3 } + 1$ 
D)  $343 p ^ { 3 } - 1$

Question 12

Determine the degree of the polynomial.
-$t - 2 t ^ { 2 } + 5 t ^ { 3 } + 1$

Accepted Answer

A)  6 
B)  1 
C)  4 
D)  3 
A)  6 
B)  1 
C)  4 
D)  3

Question 13

Factor by any method.
-$$( r + s ) ^ { 2 } - 9$$

Accepted Answer

A)  $\left( r ^ { 2 } + 3 \right) \left( s ^ { 2 } - 3 \right)$ 
B)  $( r + s + 3 ) ( r + s - 3 )$ 
C)  $( \mathrm { r } + \mathrm { s } + 3 ) ( \mathrm { r } - \mathrm { s } - 3 )$ 
D)  $( r - s + 3 ) ( r - s - 3 )$ 
A)  $\left( r ^ { 2 } + 3 \right) \left( s ^ { 2 } - 3 \right)$ 
B)  $( r + s + 3 ) ( r + s - 3 )$ 
C)  $( \mathrm { r } + \mathrm { s } + 3 ) ( \mathrm { r } - \mathrm { s } - 3 )$ 
D)  $( r - s + 3 ) ( r - s - 3 )$

Question 14

Use the distributive property to rewrite a sum as a product or the product as a sum.
-6 - 8b

Accepted Answer

A)  2(3 - 7b) 
B) 14b 
C)  2(3 - 4b) 
D) -2b 
A)  2(3 - 7b) 
B) 14b 
C)  2(3 - 4b) 
D) -2b

Question 15

Factor, using the given factor. Assume all variables represent positive real numbers.
-$( 3 m + 5 ) ^ { - 2 / 5 } + ( 3 m + 5 ) ^ { 1 / 5 } + ( 3 m + 5 ) ^ { 7 / 5 } ; \quad ( 3 m + 5 ) ^ { - 2 / 5 }$

Accepted Answer

A)  $( 3 m + 5 ) ^ { - 2 / 5 } \left[ 1 + ( 3 m + 5 ) ^ { 3 / 5 } + ( 3 m + 5 ) ^ { 9 / 5 } \right]$ 
B)  $( 3 m + 5 ) ^ { - 2 / 5 } \left[ 1 + ( 3 m + 5 ) ^ { - 3 / 5 } + ( 3 m + 5 ) ^ { - 9 / 5 } \right]$ 
C)  $( 3 m + 5 ) ^ { - 2 / 5 } \left[ 1 + ( 3 m + 5 ) ^ { 3 / 5 } - ( 3 m + 5 ) ^ { 9 / 5 } \right]$ 
D)  $( 3 m + 5 ) ^ { - 2 / 5 } \left[ 1 + ( 3 m + 5 ) ^ { 4 / 5 } + ( 3 m + 5 ) ^ { 7 / 5 } \right]$ 
A)  $( 3 m + 5 ) ^ { - 2 / 5 } \left[ 1 + ( 3 m + 5 ) ^ { 3 / 5 } + ( 3 m + 5 ) ^ { 9 / 5 } \right]$ 
B)  $( 3 m + 5 ) ^ { - 2 / 5 } \left[ 1 + ( 3 m + 5 ) ^ { - 3 / 5 } + ( 3 m + 5 ) ^ { - 9 / 5 } \right]$ 
C)  $( 3 m + 5 ) ^ { - 2 / 5 } \left[ 1 + ( 3 m + 5 ) ^ { 3 / 5 } - ( 3 m + 5 ) ^ { 9 / 5 } \right]$ 
D)  $( 3 m + 5 ) ^ { - 2 / 5 } \left[ 1 + ( 3 m + 5 ) ^ { 4 / 5 } + ( 3 m + 5 ) ^ { 7 / 5 } \right]$

Question 16

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

Accepted Answer

A)  $\frac { 27 - 10 \sqrt { 2 } } { 23 }$ 
B)  $- 1$ 
C)  $\frac { 23 - 10 \sqrt { 2 } } { 27 }$ 
D)  $\frac { 27 + 10 \sqrt { 2 } } { 23 }$ 
A)  $\frac { 27 - 10 \sqrt { 2 } } { 23 }$ 
B)  $- 1$ 
C)  $\frac { 23 - 10 \sqrt { 2 } } { 27 }$ 
D)  $\frac { 27 + 10 \sqrt { 2 } } { 23 }$

Question 17

Find the product.
-$$( n - 9 ) ( n + 9 )$$

Accepted Answer

A)  $n ^ { 2 } - 18$ 
B)  $n ^ { 2 } - 81$ 
C)  $n ^ { 2 } - 18 n - 81$ 
D)  $n ^ { 2 } - 18 n + 81$ 
A)  $n ^ { 2 } - 18$ 
B)  $n ^ { 2 } - 81$ 
C)  $n ^ { 2 } - 18 n - 81$ 
D)  $n ^ { 2 } - 18 n + 81$

Question 18

Write the expression with only positive exponents and evaluate if possible. Assume all variables represent nonzero real
numbers.
-$$\frac { \frac { 7 } { p ^ { 2 } - 16 } + p } { \frac { 1 } { p - 4 } }$$

Accepted Answer

A)  $\frac { 7 + p ^ { 2 } - 16 p } { p + 16 }$ 
B)  $\frac { 7 + p ^ { 3 } - 16 p } { p + 4 }$ 
C)  $\frac { 1 } { p ^ { 2 } - 4 }$ 
D)  $\frac { 7 + p ^ { 3 } + 16 p } { p }$ 
A)  $\frac { 7 + p ^ { 2 } - 16 p } { p + 16 }$ 
B)  $\frac { 7 + p ^ { 3 } - 16 p } { p + 4 }$ 
C)  $\frac { 1 } { p ^ { 2 } - 4 }$ 
D)  $\frac { 7 + p ^ { 3 } + 16 p } { p }$

Question 19

Write the expression with only positive exponents and evaluate if possible. Assume all variables represent nonzero real
numbers.
-$$- ( - 2 ) ^ { - 3 }$$

Accepted Answer

A)  $- 8$ 
B)  $- \frac { 1 } { 8 }$ 
C)  $\frac { 1 } { 8 }$ 
D)  8 
A)  $- 8$ 
B)  $- \frac { 1 } { 8 }$ 
C)  $\frac { 1 } { 8 }$ 
D)  8

Question 20

Use these sets to find the following. Identify any disjoint sets.
-Let $\mathrm { U } = \{ 7,8,9,10,11,12,13,14,15,16,17 \} , \mathrm { M } = \{ 7,9,11,13 \} , \mathrm { N } = \{ 8,10,12,14,16 \}$, $\mathrm { Q } = \{ 7,9,11,13,15,17 \}$, and $\mathrm { R } = \{ 7,8,9,10 \}$.
$Q \cap R ^ { \prime }$

Accepted Answer

A)  $\{ 11,13,15,17 \}$ 
B)  $\varnothing ; Q$ and $R ^ { \prime }$ are disjoint sets. 
C)  $\{ 9,11,13,15,17 \}$ 
D)  $\{ 10,11,12,13,14,15,16,17 \}$ 
A)  $\{ 11,13,15,17 \}$ 
B)  $\varnothing ; Q$ and $R ^ { \prime }$ are disjoint sets. 
C)  $\{ 9,11,13,15,17 \}$ 
D)  $\{ 10,11,12,13,14,15,16,17 \}$

Factor by grouping. - $15 a ^ { 3 } + 9 a ^ { 2 } b + 25 a b ^ { 2 } + 15 b ^ { 3 }$

Evaluate the expression. --2[6 - 6(-9 - 6)]

Simplify the expression. Assume all variables represent positive real numbers. - $\sqrt [ 3 ] { \frac { 8 } { 9 x ^ { 2 } } }$

Simplify the expression. Assume all variables represent positive real numbers. - $- \sqrt [ 4 ] { 625 }$

Factor the polynomial by substitution. - $9 y ^ { 4 } + 6 y ^ { 2 } - 8$

Identify the property illustrated by the statement. Assume all variables represent real numbers. -9(x + 4) = 9x + 36

Perform the indicated operations. Write the result using only positive exponents. Assume all variables represent nonzero real numbers. - $\frac { x ^ { - 2 } } { x ^ { - 15 } }$

Solve the problem. -If the lengths of the sides of a square are tripled, by what factor will the area change?

Identify the property illustrated by the statement. Assume all variables represent real numbers. -(9 + 1) + 1 = (1 + 9) + 1

Simplify the expression. Assume all variables represent positive real numbers. - $- \sqrt { 147 }$

Find the product. - $( 7 p - 1 ) \left( 49 p ^ { 2 } + 7 p + 1 \right)$

Determine the degree of the polynomial. - $t - 2 t ^ { 2 } + 5 t ^ { 3 } + 1$

Factor by any method. - $( r + s ) ^ { 2 } - 9$

Use the distributive property to rewrite a sum as a product or the product as a sum. -6 - 8b

Factor, using the given factor. Assume all variables represent positive real numbers. - $( 3 m + 5 ) ^ { - 2 / 5 } + ( 3 m + 5 ) ^ { 1 / 5 } + ( 3 m + 5 ) ^ { 7 / 5 } ; \quad ( 3 m + 5 ) ^ { - 2 / 5 }$

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

Find the product. - $( n - 9 ) ( n + 9 )$

Write the expression with only positive exponents and evaluate if possible. Assume all variables represent nonzero real numbers. - $\frac { \frac { 7 } { p ^ { 2 } - 16 } + p } { \frac { 1 } { p - 4 } }$

Write the expression with only positive exponents and evaluate if possible. Assume all variables represent nonzero real numbers. - $- ( - 2 ) ^ { - 3 }$

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Exam 12: Review of Basic Concepts

Factor by grouping. - 15a3+9a2b+25ab2+15b315 a ^ { 3 } + 9 a ^ { 2 } b + 25 a b ^ { 2 } + 15 b ^ { 3 }15a3+9a2b+25ab2+15b3