Question 1

Divide.
-$\frac { 3 x ^ { 2 } } { 5 } \div \frac { x ^ { 3 } } { 20 }$

Accepted Answer

A)  $\frac { x } { 12 }$ 
B)  $\frac { 60 x ^ { 2 } } { 5 x ^ { 3 } }$ 
C)  $\frac { 12 x ^ { 2 } } { x ^ { 3 } }$ 
D)  $\frac { 12 } { x }$ 
A)  $\frac { x } { 12 }$ 
B)  $\frac { 60 x ^ { 2 } } { 5 x ^ { 3 } }$ 
C)  $\frac { 12 x ^ { 2 } } { x ^ { 3 } }$ 
D)  $\frac { 12 } { x }$

Question 2

Multiply.
-$\frac { x - 8 } { x - 1 } \cdot \frac { 1 - x } { 8 - x }$

Accepted Answer

A) - 8 
B) 8 
C) -1 
D) 1 
A) - 8 
B) 8 
C) -1 
D) 1

Question 3

Simplify the rational expression.
-$\frac { 3 x ^ { 2 } - 27 } { 18 - 6 x }$

Accepted Answer

A)  $\frac { x ^ { 2 } - 9 } { 3 - x }$ 
B)  $- \frac { x + 3 } { 2 }$ 
C)  $\frac { x + 3 } { 2 }$ 
D)  $- 2 ( x + 3 )$ 
A)  $\frac { x ^ { 2 } - 9 } { 3 - x }$ 
B)  $- \frac { x + 3 } { 2 }$ 
C)  $\frac { x + 3 } { 2 }$ 
D)  $- 2 ( x + 3 )$

Question 4

Divide.
-$\frac { x ^ { 2 } + 5 x + 6 } { x ^ { 2 } + 11 x + 18 } \div \frac { x ^ { 2 } + 3 x } { x ^ { 2 } + 18 x + 81 }$

Accepted Answer

A)  $\frac { x + 9 } { x ^ { 2 } + 9 x }$ 
B)  $x + 9$ 
C)  $\frac { x } { x ^ { 2 } + 11 x + 18 }$ 
D)  $\frac { x + 9 } { x }$ 
A)  $\frac { x + 9 } { x ^ { 2 } + 9 x }$ 
B)  $x + 9$ 
C)  $\frac { x } { x ^ { 2 } + 11 x + 18 }$ 
D)  $\frac { x + 9 } { x }$

Question 5

Multiply.
-$\frac { 2 z ^ { 3 } } { 3 } \cdot \frac { 18 } { z ^ { 2 } }$

Accepted Answer

A)  $12 \mathrm { z }$ 
B)  $\frac { 12 } { \mathrm { z } }$ 
C)  $\frac { 12 z ^ { 2 } } { z ^ { 3 } }$ 
D)  $\frac { \mathrm { z } } { 12 }$ 
A)  $12 \mathrm { z }$ 
B)  $\frac { 12 } { \mathrm { z } }$ 
C)  $\frac { 12 z ^ { 2 } } { z ^ { 3 } }$ 
D)  $\frac { \mathrm { z } } { 12 }$

Question 6

Simplify the rational expression.
-$\frac { x ^ { 2 } + 8 x + 16 } { x ^ { 2 } + 12 x + 32 }$

Accepted Answer

A)  $\frac { 8 x + 1 } { 12 x + 2 }$ 
B) 128x x ++ 16 32 
C) x x ++ 4 8 
D) - xx22++ 12 8x x ++ 16 32 
A)  $\frac { 8 x + 1 } { 12 x + 2 }$ 
B) 128x x ++ 16 32 
C) x x ++ 4 8 
D) - xx22++ 12 8x x ++ 16 32

Question 7

Simplify the rational expression.
-$\frac { 5 - m } { m - 5 }$

Accepted Answer

A) 1 
B) -m 
C) -1 
D) 0 
A) 1 
B) -m 
C) -1 
D) 0

Question 8

Add or subtract.
-$\frac { 7 x - 7 } { x } + \frac { 8 x - 7 } { 5 x }$

Accepted Answer

A)  $\frac { 43 x + 28 } { 5 x }$ 
B)  $\frac { 43 x - 42 } { 5 x }$ 
C)  $\frac { 43 x - 42 } { 5 x ^ { 2 } }$ 
D)  $\frac { 27 x - 42 } { 5 x }$ 
A)  $\frac { 43 x + 28 } { 5 x }$ 
B)  $\frac { 43 x - 42 } { 5 x }$ 
C)  $\frac { 43 x - 42 } { 5 x ^ { 2 } }$ 
D)  $\frac { 27 x - 42 } { 5 x }$

Question 9

Simplify the rational expression.
-$\frac { q ^ { 3 } + 216 } { 2 q ^ { 2 } + 15 + 18 }$

Accepted Answer

A)  $\frac { q ^ { 2 } - 6 q + 36 } { 2 q - 3 }$ 
B)  $\frac { \mathrm { q } ^ { 2 } + 6 \mathrm { q } + 36 } { 2 \mathrm { q } - 3 }$ 
C)  $\frac { q ^ { 2 } - 6 q + 36 } { 2 q + 3 }$ 
D)  $\frac { q ^ { 2 } + 6 q - 36 } { 2 q + 3 }$ 
A)  $\frac { q ^ { 2 } - 6 q + 36 } { 2 q - 3 }$ 
B)  $\frac { \mathrm { q } ^ { 2 } + 6 \mathrm { q } + 36 } { 2 \mathrm { q } - 3 }$ 
C)  $\frac { q ^ { 2 } - 6 q + 36 } { 2 q + 3 }$ 
D)  $\frac { q ^ { 2 } + 6 q - 36 } { 2 q + 3 }$

Question 10

Find the least common denominator.
-$\frac { 8 } { 20 x y } + \frac { 7 } { 16 x ^ { 2 } }$

Accepted Answer

A)  $80 x ^ { 2 } y$ 
B)  $80 x y ^ { 2 }$ 
C)  $80 x ^ { 3 } y$ 
D)  $80 x y ^ { 3 }$ 
A)  $80 x ^ { 2 } y$ 
B)  $80 x y ^ { 2 }$ 
C)  $80 x ^ { 3 } y$ 
D)  $80 x y ^ { 3 }$

Question 11

Add or subtract.
-$\frac { x } { x + 4 } + \frac { 3 } { x + 4 }$

Accepted Answer

A)  $\frac { 3 } { 4 }$ 
B)  $\frac { x + 3 } { x + 4 }$ 
C)  $\frac { 4 x } { x + 4 }$ 
D)  $\frac { 3 x } { x ^ { 2 } + 8 x + 16 }$ 
A)  $\frac { 3 } { 4 }$ 
B)  $\frac { x + 3 } { x + 4 }$ 
C)  $\frac { 4 x } { x + 4 }$ 
D)  $\frac { 3 x } { x ^ { 2 } + 8 x + 16 }$

Question 12

Simplify the rational expression.
-$\frac { m ^ { 2 } - 10 m n + 24 n ^ { 2 } } { m ^ { 2 } - 8 m n + 16 n ^ { 2 } }$

Accepted Answer

A)  $\frac { m - 6 n } { ( m - 4 n ) ^ { 2 } }$ 
B)  $\frac { m - 4 n } { m - 6 n }$ 
C)  $\frac { m - 6 n } { m - 4 n }$ 
D)  $\frac { m - 6 } { m - 4 }$ 
A)  $\frac { m - 6 n } { ( m - 4 n ) ^ { 2 } }$ 
B)  $\frac { m - 4 n } { m - 6 n }$ 
C)  $\frac { m - 6 n } { m - 4 n }$ 
D)  $\frac { m - 6 } { m - 4 }$

Question 13

Simplify the rational expression.
-$\frac { y ^ { 2 } + 5 y - 24 } { y ^ { 2 } + 3 y - 40 }$

Accepted Answer

A)  $\frac { 5 y - 24 } { 3 y - 40 }$ 
B) - y22 + 5 3y - 24 40 
C) 53y - 3 5 
D) y - 3 5 
A)  $\frac { 5 y - 24 } { 3 y - 40 }$ 
B) - y22 + 5 3y - 24 40 
C) 53y - 3 5 
D) y - 3 5

Question 14

Determine the value or values that are excluded from the following expression.
-$$\frac { 2 } { x - 3 }$$

Accepted Answer

A) -3 
B) 0 
C) 3 
D) none 
A) -3 
B) 0 
C) 3 
D) none

Question 15

Determine the domain of the function.
-$f ( x ) = \frac { 2 x } { x - 9 }$

Accepted Answer

A)  $\{ x \mid x \neq - 9 \}$ 
B)  $\{ x \mid x \neq 0 \}$ 
C)  $\{ x \mid x \text { is a real number } \}$ 
D)  $\{ x \mid x \neq 9 \}$ 
A)  $\{ x \mid x \neq - 9 \}$ 
B)  $\{ x \mid x \neq 0 \}$ 
C)  $\{ x \mid x \text { is a real number } \}$ 
D)  $\{ x \mid x \neq 9 \}$

Question 16

Determine the value or values that are excluded from the following expression.
-$\frac { x - 36 } { x ^ { 2 } - 7 x + 10 }$

Accepted Answer

A) 5, 2 
B) -5, -2 
C) 6 
D) 0 
A) 5, 2 
B) -5, -2 
C) 6 
D) 0

Question 17

Add or subtract.
-$\frac { 11 y ^ { 2 } + 28 y + 27 } { 25 y ^ { 2 } - 64 } - \frac { 6 y ^ { 2 } - 5 } { 25 y ^ { 2 } - 64 }$

Accepted Answer

A)  $\frac { 5 y - 8 } { 5 y + 8 }$ 
B)  $\frac { 11 y + 1 } { 5 y + 8 }$ 
C)  $\frac { y + 4 } { 5 y - 8 }$ 
D)  $\frac { 5 y + 8 } { 5 y - 8 }$ 
A)  $\frac { 5 y - 8 } { 5 y + 8 }$ 
B)  $\frac { 11 y + 1 } { 5 y + 8 }$ 
C)  $\frac { y + 4 } { 5 y - 8 }$ 
D)  $\frac { 5 y + 8 } { 5 y - 8 }$

Question 18

Find the least common denominator.
-$$\frac { 3 x } { x ^ { 2 } - 5 x - 36 } - \frac { x - 1 } { x ^ { 2 } - 16 } + \frac { 1 } { x ^ { 2 } - 13 x + 36 }$$

Accepted Answer

A)  $( x - 9 ) ( x + 4 ) ^ { 2 }$ 
B) (x - 9)(x + 4) 
C) (x - 9)(x + 4)(x - 4) 
D) (x - 9)(x - 4) 
A)  $( x - 9 ) ( x + 4 ) ^ { 2 }$ 
B) (x - 9)(x + 4) 
C) (x - 9)(x + 4)(x - 4) 
D) (x - 9)(x - 4)

Question 19

Solve the problem.
-Consider the rectangle below. Its area is $$x ^ { 2 } + 2 x y + y ^ { 2 }$$ and its width is 17x + 17y. Find its width, w, in terms of x and by dividing its area by its length.

Accepted Answer

A)  $\frac { x + 2 x y + y } { 17 }$ 
B)  $\frac { 17 } { x + y }$ 
C)  $\frac { x + y } { 17 }$ 
D)  $\frac { x ^ { 2 } + 2 x y + y ^ { 2 } } { 17 ( x + y ) }$ 
A)  $\frac { x + 2 x y + y } { 17 }$ 
B)  $\frac { 17 } { x + y }$ 
C)  $\frac { x + y } { 17 }$ 
D)  $\frac { x ^ { 2 } + 2 x y + y ^ { 2 } } { 17 ( x + y ) }$

Question 20

Divide.
-$\frac { 5 b + 10 } { 4 a + 20 } \div \frac { a b - 5 b + 2 a - 10 } { a ^ { 2 } - 10 a + 25 }$

Accepted Answer

A)  $\frac { 4 ( a + 5 ) } { 5 ( a - 5 ) }$ 
B)  $\frac { 5 ( a + 5 ) } { 4 ( a - 5 ) }$ 
C)  $- \frac { 5 ( a - 5 ) } { 4 ( a + 5 ) }$ 
D)  $\frac { 5 ( a - 5 ) } { 4 ( a + 5 ) }$ 
A)  $\frac { 4 ( a + 5 ) } { 5 ( a - 5 ) }$ 
B)  $\frac { 5 ( a + 5 ) } { 4 ( a - 5 ) }$ 
C)  $- \frac { 5 ( a - 5 ) } { 4 ( a + 5 ) }$ 
D)  $\frac { 5 ( a - 5 ) } { 4 ( a + 5 ) }$

Divide. - $\frac { 3 x ^ { 2 } } { 5 } \div \frac { x ^ { 3 } } { 20 }$

Multiply. - $\frac { x - 8 } { x - 1 } \cdot \frac { 1 - x } { 8 - x }$

Simplify the rational expression. - $\frac { 3 x ^ { 2 } - 27 } { 18 - 6 x }$

Divide. - $\frac { x ^ { 2 } + 5 x + 6 } { x ^ { 2 } + 11 x + 18 } \div \frac { x ^ { 2 } + 3 x } { x ^ { 2 } + 18 x + 81 }$

Multiply. - $\frac { 2 z ^ { 3 } } { 3 } \cdot \frac { 18 } { z ^ { 2 } }$

Simplify the rational expression. - $\frac { x ^ { 2 } + 8 x + 16 } { x ^ { 2 } + 12 x + 32 }$

Simplify the rational expression. - $\frac { 5 - m } { m - 5 }$

Add or subtract. - $\frac { 7 x - 7 } { x } + \frac { 8 x - 7 } { 5 x }$

Simplify the rational expression. - $\frac { q ^ { 3 } + 216 } { 2 q ^ { 2 } + 15 + 18 }$

Find the least common denominator. - $\frac { 8 } { 20 x y } + \frac { 7 } { 16 x ^ { 2 } }$

Add or subtract. - $\frac { x } { x + 4 } + \frac { 3 } { x + 4 }$

Simplify the rational expression. - $\frac { m ^ { 2 } - 10 m n + 24 n ^ { 2 } } { m ^ { 2 } - 8 m n + 16 n ^ { 2 } }$

Simplify the rational expression. - $\frac { y ^ { 2 } + 5 y - 24 } { y ^ { 2 } + 3 y - 40 }$

Determine the value or values that are excluded from the following expression. - $\frac { 2 } { x - 3 }$

Determine the domain of the function. - $f ( x ) = \frac { 2 x } { x - 9 }$

Determine the value or values that are excluded from the following expression. - $\frac { x - 36 } { x ^ { 2 } - 7 x + 10 }$

Add or subtract. - $\frac { 11 y ^ { 2 } + 28 y + 27 } { 25 y ^ { 2 } - 64 } - \frac { 6 y ^ { 2 } - 5 } { 25 y ^ { 2 } - 64 }$

Find the least common denominator. - $\frac { 3 x } { x ^ { 2 } - 5 x - 36 } - \frac { x - 1 } { x ^ { 2 } - 16 } + \frac { 1 } { x ^ { 2 } - 13 x + 36 }$

Divide. - $\frac { 5 b + 10 } { 4 a + 20 } \div \frac { a b - 5 b + 2 a - 10 } { a ^ { 2 } - 10 a + 25 }$

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

Divide. - 3x25÷x320\frac { 3 x ^ { 2 } } { 5 } \div \frac { x ^ { 3 } } { 20 }53x2​÷20x3​

Multiply. - x−8x−1⋅1−x8−x\frac { x - 8 } { x - 1 } \cdot \frac { 1 - x } { 8 - x }x−1x−8​⋅8−x1−x​

Simplify the rational expression. - 3x2−2718−6x\frac { 3 x ^ { 2 } - 27 } { 18 - 6 x }18−6x3x2−27​

Divide. - x2+5x+6x2+11x+18÷x2+3xx2+18x+81\frac { x ^ { 2 } + 5 x + 6 } { x ^ { 2 } + 11 x + 18 } \div \frac { x ^ { 2 } + 3 x } { x ^ { 2 } + 18 x + 81 }x2+11x+18x2+5x+6​÷x2+18x+81x2+3x​

Multiply. - 2z33⋅18z2\frac { 2 z ^ { 3 } } { 3 } \cdot \frac { 18 } { z ^ { 2 } }32z3​⋅z218​

Simplify the rational expression. - x2+8x+16x2+12x+32\frac { x ^ { 2 } + 8 x + 16 } { x ^ { 2 } + 12 x + 32 }x2+12x+32x2+8x+16​

Simplify the rational expression. - 5−mm−5\frac { 5 - m } { m - 5 }m−55−m​

Add or subtract. - 7x−7x+8x−75x\frac { 7 x - 7 } { x } + \frac { 8 x - 7 } { 5 x }x7x−7​+5x8x−7​

Simplify the rational expression. - q3+2162q2+15+18\frac { q ^ { 3 } + 216 } { 2 q ^ { 2 } + 15 + 18 }2q2+15+18q3+216​

Find the least common denominator. - 820xy+716x2\frac { 8 } { 20 x y } + \frac { 7 } { 16 x ^ { 2 } }20xy8​+16x27​

Add or subtract. - xx+4+3x+4\frac { x } { x + 4 } + \frac { 3 } { x + 4 }x+4x​+x+43​

Simplify the rational expression. - m2−10mn+24n2m2−8mn+16n2\frac { m ^ { 2 } - 10 m n + 24 n ^ { 2 } } { m ^ { 2 } - 8 m n + 16 n ^ { 2 } }m2−8mn+16n2m2−10mn+24n2​

Simplify the rational expression. - y2+5y−24y2+3y−40\frac { y ^ { 2 } + 5 y - 24 } { y ^ { 2 } + 3 y - 40 }y2+3y−40y2+5y−24​

Determine the value or values that are excluded from the following expression. - 2x−3\frac { 2 } { x - 3 }x−32​

Determine the domain of the function. - f(x)=2xx−9f ( x ) = \frac { 2 x } { x - 9 }f(x)=x−92x​

Determine the value or values that are excluded from the following expression. - x−36x2−7x+10\frac { x - 36 } { x ^ { 2 } - 7 x + 10 }x2−7x+10x−36​

Add or subtract. - 11y2+28y+2725y2−64−6y2−525y2−64\frac { 11 y ^ { 2 } + 28 y + 27 } { 25 y ^ { 2 } - 64 } - \frac { 6 y ^ { 2 } - 5 } { 25 y ^ { 2 } - 64 }25y2−6411y2+28y+27​−25y2−646y2−5​

Find the least common denominator. - 3xx2−5x−36−x−1x2−16+1x2−13x+36\frac { 3 x } { x ^ { 2 } - 5 x - 36 } - \frac { x - 1 } { x ^ { 2 } - 16 } + \frac { 1 } { x ^ { 2 } - 13 x + 36 }x2−5x−363x​−x2−16x−1​+x2−13x+361​

Solve the problem. -Consider the rectangle below. Its area is x2+2xy+y2x ^ { 2 } + 2 x y + y ^ { 2 }x2+2xy+y2 and its width is 17x + 17y. Find its width, w, in terms of x and by dividing its area by its length.

Divide. - 5b+104a+20÷ab−5b+2a−10a2−10a+25\frac { 5 b + 10 } { 4 a + 20 } \div \frac { a b - 5 b + 2 a - 10 } { a ^ { 2 } - 10 a + 25 }4a+205b+10​÷a2−10a+25ab−5b+2a−10​