Question 1

Find the least common denominator.
-$\frac { 8 } { t } + \frac { 7 } { t + 7 }$

Accepted Answer

A) 56 
B) t(t + 7) 
C) t + 7 
D) t 
A) 56 
B) t(t + 7) 
C) t + 7 
D) t

Question 2

Simplify the rational expression.
-$\frac { 9 x ^ { 2 } - 79 x + 56 } { x - 8 }$

Accepted Answer

A)  $\frac { 1 } { x - 8 }$ 
B)  $9 x - 7$ 
C)  $\frac { 9 x ^ { 2 } - 79 x + 56 } { x - 8 }$ 
D)  $9 x ^ { 2 } - 86$ 
A)  $\frac { 1 } { x - 8 }$ 
B)  $9 x - 7$ 
C)  $\frac { 9 x ^ { 2 } - 79 x + 56 } { x - 8 }$ 
D)  $9 x ^ { 2 } - 86$

Question 3

Determine the value or values that are excluded from the following expression.
-$\frac { x + 5 } { x ^ { 2 } + 36 }$

Accepted Answer

A) -6, -5, 6 
B) -6, 6 
C) 0, -36 
D) none 
A) -6, -5, 6 
B) -6, 6 
C) 0, -36 
D) none

Question 4

Find the least common denominator.
-$\frac { 2 } { 9 x ^ { 4 } } + \frac { 8 } { 27 x ^ { 2 } }$

Accepted Answer

A)  $27 x ^ { 6 }$ 
B)  $243 x ^ { 6 }$ 
C)  $27 x ^ { 4 }$ 
D)  $243 x ^ { 4 }$ 
A)  $27 x ^ { 6 }$ 
B)  $243 x ^ { 6 }$ 
C)  $27 x ^ { 4 }$ 
D)  $243 x ^ { 4 }$

Question 5

Determine the value or values that are excluded from the following expression.
-$\frac { x ^ { 2 } - 25 } { 2 x ^ { 2 } - 5 x - 7 }$

Accepted Answer

A) -5, 5 
B)  $- 5 , - 1 , \frac { 7 } { 2 } , 5$ 
C)  $- \frac { 7 } { 2 } , 1$ 
D)  $- 1 , \frac { 7 } { 2 }$ 
A) -5, 5 
B)  $- 5 , - 1 , \frac { 7 } { 2 } , 5$ 
C)  $- \frac { 7 } { 2 } , 1$ 
D)  $- 1 , \frac { 7 } { 2 }$

Question 6

Add or subtract.
-$- \frac { 5 } { 14 } - \frac { 6 } { 7 x }$

Accepted Answer

A)  $\frac { - 5 x + 12 } { 14 x }$ 
B)  $\frac { - 5 x - 12 } { 14 x }$ 
C)  $\frac { - 11 } { 14 - 7 x }$ 
D)  $\frac { 5 x - 12 } { 14 x }$ 
A)  $\frac { - 5 x + 12 } { 14 x }$ 
B)  $\frac { - 5 x - 12 } { 14 x }$ 
C)  $\frac { - 11 } { 14 - 7 x }$ 
D)  $\frac { 5 x - 12 } { 14 x }$

Question 7

Add or subtract.
-$\frac { 4 } { x + 5 } - \frac { 2 } { x - 5 }$

Accepted Answer

A)  $\frac { 2 x - 10 } { ( x + 5 ) ( x - 5 ) }$ 
B)  $\frac { 2 } { ( x + 5 ) ( x - 5 ) }$ 
C)  $\frac { 2 x - 30 } { ( x + 5 ) ( x - 5 ) }$ 
D)  $\frac { 2 x + 30 } { ( x + 5 ) ( x - 5 ) }$ 
A)  $\frac { 2 x - 10 } { ( x + 5 ) ( x - 5 ) }$ 
B)  $\frac { 2 } { ( x + 5 ) ( x - 5 ) }$ 
C)  $\frac { 2 x - 30 } { ( x + 5 ) ( x - 5 ) }$ 
D)  $\frac { 2 x + 30 } { ( x + 5 ) ( x - 5 ) }$

Question 8

Find the least common denominator.
-$\frac { 1 } { r ^ { 2 } + 10 r + 25 } - \frac { 1 } { r ^ { 2 } + 5 r }$

Accepted Answer

A)  $( r + 5 ) ^ { 2 }$ 
B)  $r ( r + 5 ) ^ { 2 }$ 
C) r(r + 1)(r + 5) 
D) r(r + 5) 
A)  $( r + 5 ) ^ { 2 }$ 
B)  $r ( r + 5 ) ^ { 2 }$ 
C) r(r + 1)(r + 5) 
D) r(r + 5)

Question 9

Simplify the rational expression.
-$\frac { m ^ { 2 } - 9 m } { 9 - m }$

Accepted Answer

A) -(m + 3) 
B) m + 3 
C) -m 
D) m 
A) -(m + 3) 
B) m + 3 
C) -m 
D) m

Question 10

Multiply.
-$\frac { ( a + b ) ^ { 3 } } { a ^ { 3 } + b ^ { 3 } } \cdot \frac { a ^ { 2 } - b ^ { 2 } } { ( a + b ) ^ { 2 } }$

Accepted Answer

A)  $\frac { a ^ { 2 } - b ^ { 2 } } { a ^ { 2 } + b ^ { 2 } }$ 
B) -1 
C)  $\frac { ( a + b ) ( a - b ) } { a ^ { 2 } - a b + b ^ { 2 } }$ 
D)  $\frac { ( a + b ) ( a - b ) } { a ^ { 2 } + a b + b ^ { 2 } }$ 
A)  $\frac { a ^ { 2 } - b ^ { 2 } } { a ^ { 2 } + b ^ { 2 } }$ 
B) -1 
C)  $\frac { ( a + b ) ( a - b ) } { a ^ { 2 } - a b + b ^ { 2 } }$ 
D)  $\frac { ( a + b ) ( a - b ) } { a ^ { 2 } + a b + b ^ { 2 } }$

Question 11

Divide.
-$\frac { z ^ { 2 } + 11 z + 30 } { z ^ { 2 } + 14 z + 48 } \div \frac { z ^ { 2 } + 5 z } { z ^ { 2 } + 12 z + 32 }$

Accepted Answer

A)  $\frac { z + 4 } { z ^ { 2 } + 8 z }$ 
B)  $\frac { z + 4 } { z }$ 
C)  $\frac { z } { z ^ { 2 } + 14 z + 48 }$ 
D) z + 4 
A)  $\frac { z + 4 } { z ^ { 2 } + 8 z }$ 
B)  $\frac { z + 4 } { z }$ 
C)  $\frac { z } { z ^ { 2 } + 14 z + 48 }$ 
D) z + 4

Question 12

Simplify the rational expression.
-$\frac { \mathbf { r } + \mathbf { r s } } { \mathrm { r } }$

Accepted Answer

A)  $1 + \mathrm { rs }$ 
B) s 
C) 1 + s 
D) r + s 
A)  $1 + \mathrm { rs }$ 
B) s 
C) 1 + s 
D) r + s

Question 13

Add or subtract.
-$\frac { 3 } { x ^ { 2 } - 3 x + 2 } + \frac { 7 } { x ^ { 2 } - 1 }$

Accepted Answer

A)  $\frac { 10 x - 11 } { ( x - 1 ) ( x + 1 ) ( x - 2 ) }$ 
B)  $\frac { 11 x - 10 } { ( x - 1 ) ( x + 1 ) ( x - 2 ) }$ 
C)  $\frac { 42 x - 11 } { ( x - 1 ) ( x + 1 ) ( x - 2 ) }$ 
D)  $\frac { 10 x - 11 } { ( x - 1 ) ( x - 2 ) }$ 
A)  $\frac { 10 x - 11 } { ( x - 1 ) ( x + 1 ) ( x - 2 ) }$ 
B)  $\frac { 11 x - 10 } { ( x - 1 ) ( x + 1 ) ( x - 2 ) }$ 
C)  $\frac { 42 x - 11 } { ( x - 1 ) ( x + 1 ) ( x - 2 ) }$ 
D)  $\frac { 10 x - 11 } { ( x - 1 ) ( x - 2 ) }$

Question 14

Determine the domain of the function.
-$g ( x ) = \frac { x + 8 } { x ^ { 2 } + 25 x }$

Accepted Answer

A)  $\{ x \mid x \neq - 5 , x \neq 5 \text {, and } x \neq - 8 \}$ 
B)  $\{ x \mid x \neq - 5 \text { and } x \neq 5 \}$ 
C)  $\{ x \mid x \neq 0 \text { and } x \neq - 25 \}$ 
D)  $\{ x \mid x \text { is a real number } \}$ 
A)  $\{ x \mid x \neq - 5 , x \neq 5 \text {, and } x \neq - 8 \}$ 
B)  $\{ x \mid x \neq - 5 \text { and } x \neq 5 \}$ 
C)  $\{ x \mid x \neq 0 \text { and } x \neq - 25 \}$ 
D)  $\{ x \mid x \text { is a real number } \}$

Question 15

Add or subtract.
-$\frac { x } { x ^ { 2 } - 16 } - \frac { 3 } { x ^ { 2 } + 5 x + 4 }$

Accepted Answer

A)  $\frac { x ^ { 2 } + 2 x + 12 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
B)  $\frac { x ^ { 2 } - 2 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
C)  $\frac { x ^ { 2 } - 2 x + 12 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
D)  $\frac { x ^ { 2 } - 2 x + 12 } { ( x - 4 ) ( x + 4 ) }$ 
A)  $\frac { x ^ { 2 } + 2 x + 12 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
B)  $\frac { x ^ { 2 } - 2 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
C)  $\frac { x ^ { 2 } - 2 x + 12 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
D)  $\frac { x ^ { 2 } - 2 x + 12 } { ( x - 4 ) ( x + 4 ) }$

Question 16

Add or subtract.
-$\frac { 3 } { x + 9 } + 9$

Accepted Answer

A)  $\frac { 9 x + 108 } { x + 9 }$ 
B)  $\frac { 12 } { x + 9 }$ 
C)  $\frac { 9 x + 78 } { x + 9 }$ 
D)  $\frac { 9 x + 84 } { x + 9 }$ 
A)  $\frac { 9 x + 108 } { x + 9 }$ 
B)  $\frac { 12 } { x + 9 }$ 
C)  $\frac { 9 x + 78 } { x + 9 }$ 
D)  $\frac { 9 x + 84 } { x + 9 }$

Question 17

Find the least common denominator.
-$$\frac { 2 } { x ^ { 2 } + 3 x - 18 } - \frac { 8 } { x ^ { 2 } + 2 x - 15 }$$

Accepted Answer

A) (x + 6)(x - 3)(x + 5) 
B) (x + 6)(x - 3) 
C) (x - 6)(x + 3)(x + 5) 
D) (x - 3)(x + 5) 
A) (x + 6)(x - 3)(x + 5) 
B) (x + 6)(x - 3) 
C) (x - 6)(x + 3)(x + 5) 
D) (x - 3)(x + 5)

Question 18

Divide.
-$\frac { ( p - q ) ^ { 2 } } { ( p + q ) ^ { 2 } } \div \frac { p ^ { 2 } - 2 p q + q ^ { 2 } } { p ^ { 2 } - q ^ { 2 } }$

Accepted Answer

A)  $\frac { p ^ { 2 } - 2 p q + q ^ { 2 } } { ( p + q ) ^ { 2 } }$ 
B)  $\frac { p - q } { p + q }$ 
C)  $\frac { p + q } { p - q }$ 
D)  $- 2 p q$ 
A)  $\frac { p ^ { 2 } - 2 p q + q ^ { 2 } } { ( p + q ) ^ { 2 } }$ 
B)  $\frac { p - q } { p + q }$ 
C)  $\frac { p + q } { p - q }$ 
D)  $- 2 p q$

Question 19

Determine the value or values that are excluded from the following expression.
-$\frac { x - 5 } { x ^ { 2 } - 81 }$

Accepted Answer

A)  $\frac { 5 } { 81 }$ 
B) 9 
C) 9, -9 
D) 81 
A)  $\frac { 5 } { 81 }$ 
B) 9 
C) 9, -9 
D) 81

Question 20

Divide.
-$\frac { y ^ { 2 } + z ^ { 2 } } { y ^ { 4 } - z ^ { 4 } } \div \frac { \left( y ^ { 2 } - z ^ { 2 } \right) ^ { 2 } } { \left( y ^ { 2 } - z ^ { 2 } \right) ^ { 3 } }$

Accepted Answer

A)  $\frac { y ^ { 2 } + z ^ { 2 } } { \left( y ^ { 4 } - z ^ { 4 } \right) \left( y ^ { 2 } - z ^ { 2 } \right) }$ 
B) 1 
C)  $\frac { y ^ { 2 } - z ^ { 2 } } { y ^ { 2 } + z ^ { 2 } }$ 
D)  $\frac { y ^ { 2 } + z ^ { 2 } } { y ^ { 2 } - z ^ { 2 } }$ 
A)  $\frac { y ^ { 2 } + z ^ { 2 } } { \left( y ^ { 4 } - z ^ { 4 } \right) \left( y ^ { 2 } - z ^ { 2 } \right) }$ 
B) 1 
C)  $\frac { y ^ { 2 } - z ^ { 2 } } { y ^ { 2 } + z ^ { 2 } }$ 
D)  $\frac { y ^ { 2 } + z ^ { 2 } } { y ^ { 2 } - z ^ { 2 } }$

Find the least common denominator. - $\frac { 8 } { t } + \frac { 7 } { t + 7 }$

Simplify the rational expression. - $\frac { 9 x ^ { 2 } - 79 x + 56 } { x - 8 }$

Determine the value or values that are excluded from the following expression. - $\frac { x + 5 } { x ^ { 2 } + 36 }$

Find the least common denominator. - $\frac { 2 } { 9 x ^ { 4 } } + \frac { 8 } { 27 x ^ { 2 } }$

Determine the value or values that are excluded from the following expression. - $\frac { x ^ { 2 } - 25 } { 2 x ^ { 2 } - 5 x - 7 }$

Add or subtract. - $- \frac { 5 } { 14 } - \frac { 6 } { 7 x }$

Add or subtract. - $\frac { 4 } { x + 5 } - \frac { 2 } { x - 5 }$

Find the least common denominator. - $\frac { 1 } { r ^ { 2 } + 10 r + 25 } - \frac { 1 } { r ^ { 2 } + 5 r }$

Simplify the rational expression. - $\frac { m ^ { 2 } - 9 m } { 9 - m }$

Multiply. - $\frac { ( a + b ) ^ { 3 } } { a ^ { 3 } + b ^ { 3 } } \cdot \frac { a ^ { 2 } - b ^ { 2 } } { ( a + b ) ^ { 2 } }$

Divide. - $\frac { z ^ { 2 } + 11 z + 30 } { z ^ { 2 } + 14 z + 48 } \div \frac { z ^ { 2 } + 5 z } { z ^ { 2 } + 12 z + 32 }$

Simplify the rational expression. - $\frac { \mathbf { r } + \mathbf { r s } } { \mathrm { r } }$

Add or subtract. - $\frac { 3 } { x ^ { 2 } - 3 x + 2 } + \frac { 7 } { x ^ { 2 } - 1 }$

Determine the domain of the function. - $g ( x ) = \frac { x + 8 } { x ^ { 2 } + 25 x }$

Add or subtract. - $\frac { x } { x ^ { 2 } - 16 } - \frac { 3 } { x ^ { 2 } + 5 x + 4 }$

Add or subtract. - $\frac { 3 } { x + 9 } + 9$

Find the least common denominator. - $\frac { 2 } { x ^ { 2 } + 3 x - 18 } - \frac { 8 } { x ^ { 2 } + 2 x - 15 }$

Divide. - $\frac { ( p - q ) ^ { 2 } } { ( p + q ) ^ { 2 } } \div \frac { p ^ { 2 } - 2 p q + q ^ { 2 } } { p ^ { 2 } - q ^ { 2 } }$

Determine the value or values that are excluded from the following expression. - $\frac { x - 5 } { x ^ { 2 } - 81 }$

Divide. - $\frac { y ^ { 2 } + z ^ { 2 } } { y ^ { 4 } - z ^ { 4 } } \div \frac { \left( y ^ { 2 } - z ^ { 2 } \right) ^ { 2 } } { \left( y ^ { 2 } - z ^ { 2 } \right) ^ { 3 } }$

Basic Concepts

Equations and Inequalities

Graphs and Functions

Systems of Equations and Inequalities

Polynomials and Polynomial Functions

Roots, Radicals, and Complex Numbers

Quadratic Functions

Exponential and Logarithmic Functions

Characteristics of Functions and Their Graphs

Conic Sections

Sequences, Series, and Probability

Filters

Exam 6: Rational Expressions and Equations

Find the least common denominator. - 8t+7t+7\frac { 8 } { t } + \frac { 7 } { t + 7 }t8​+t+77​

Simplify the rational expression. - 9x2−79x+56x−8\frac { 9 x ^ { 2 } - 79 x + 56 } { x - 8 }x−89x2−79x+56​

Determine the value or values that are excluded from the following expression. - x+5x2+36\frac { x + 5 } { x ^ { 2 } + 36 }x2+36x+5​

Find the least common denominator. - 29x4+827x2\frac { 2 } { 9 x ^ { 4 } } + \frac { 8 } { 27 x ^ { 2 } }9x42​+27x28​

Determine the value or values that are excluded from the following expression. - x2−252x2−5x−7\frac { x ^ { 2 } - 25 } { 2 x ^ { 2 } - 5 x - 7 }2x2−5x−7x2−25​

Add or subtract. - −514−67x- \frac { 5 } { 14 } - \frac { 6 } { 7 x }−145​−7x6​

Add or subtract. - 4x+5−2x−5\frac { 4 } { x + 5 } - \frac { 2 } { x - 5 }x+54​−x−52​

Find the least common denominator. - 1r2+10r+25−1r2+5r\frac { 1 } { r ^ { 2 } + 10 r + 25 } - \frac { 1 } { r ^ { 2 } + 5 r }r2+10r+251​−r2+5r1​

Simplify the rational expression. - m2−9m9−m\frac { m ^ { 2 } - 9 m } { 9 - m }9−mm2−9m​

Multiply. - (a+b)3a3+b3⋅a2−b2(a+b)2\frac { ( a + b ) ^ { 3 } } { a ^ { 3 } + b ^ { 3 } } \cdot \frac { a ^ { 2 } - b ^ { 2 } } { ( a + b ) ^ { 2 } }a3+b3(a+b)3​⋅(a+b)2a2−b2​

Divide. - z2+11z+30z2+14z+48÷z2+5zz2+12z+32\frac { z ^ { 2 } + 11 z + 30 } { z ^ { 2 } + 14 z + 48 } \div \frac { z ^ { 2 } + 5 z } { z ^ { 2 } + 12 z + 32 }z2+14z+48z2+11z+30​÷z2+12z+32z2+5z​

Simplify the rational expression. - r+rsr\frac { \mathbf { r } + \mathbf { r s } } { \mathrm { r } }rr+rs​

Add or subtract. - 3x2−3x+2+7x2−1\frac { 3 } { x ^ { 2 } - 3 x + 2 } + \frac { 7 } { x ^ { 2 } - 1 }x2−3x+23​+x2−17​

Determine the domain of the function. - g(x)=x+8x2+25xg ( x ) = \frac { x + 8 } { x ^ { 2 } + 25 x }g(x)=x2+25xx+8​

Add or subtract. - xx2−16−3x2+5x+4\frac { x } { x ^ { 2 } - 16 } - \frac { 3 } { x ^ { 2 } + 5 x + 4 }x2−16x​−x2+5x+43​

Add or subtract. - 3x+9+9\frac { 3 } { x + 9 } + 9x+93​+9

Find the least common denominator. - 2x2+3x−18−8x2+2x−15\frac { 2 } { x ^ { 2 } + 3 x - 18 } - \frac { 8 } { x ^ { 2 } + 2 x - 15 }x2+3x−182​−x2+2x−158​

Divide. - (p−q)2(p+q)2÷p2−2pq+q2p2−q2\frac { ( p - q ) ^ { 2 } } { ( p + q ) ^ { 2 } } \div \frac { p ^ { 2 } - 2 p q + q ^ { 2 } } { p ^ { 2 } - q ^ { 2 } }(p+q)2(p−q)2​÷p2−q2p2−2pq+q2​

Determine the value or values that are excluded from the following expression. - x−5x2−81\frac { x - 5 } { x ^ { 2 } - 81 }x2−81x−5​

Divide. - y2+z2y4−z4÷(y2−z2)2(y2−z2)3\frac { y ^ { 2 } + z ^ { 2 } } { y ^ { 4 } - z ^ { 4 } } \div \frac { \left( y ^ { 2 } - z ^ { 2 } \right) ^ { 2 } } { \left( y ^ { 2 } - z ^ { 2 } \right) ^ { 3 } }y4−z4y2+z2​÷(y2−z2)3(y2−z2)2​