Deck 9: Rational Expressions

Question 1

Simplify the rational expression. $\frac { 25 x y } { 35 y }$

A) $\frac { 5 x y } { 7 }$
B) $\frac { 5 x } { 7 }$
C) $\frac { x } { 2 }$
D) $\frac { 1 } { 2 x }$

Answer 1

$\frac { 5 x } { 7 }$

Question 2

Simplify the rational expression. $\frac { 20 x + 16 } { 16 x - 12 }$

A) $\frac { 5 x + 16 } { 4 x - 12 }$
B) 9
C) $- \frac { 5 x + 4 } { 4 x + 3 }$
D) $\frac { 5 x + 4 } { 4 x - 3 }$

Answer 2

$\frac { 5 x + 4 } { 4 x - 3 }$

Question 3

If the indicated quotient of rational expression is $- 1$ , then the numerator and denominator are opposites.

Answer 3

True

Question 4

Simplify the rational expression. $\frac { 3 n ^ { 2 } - 14 n - 5 } { 7 n ^ { 2 } - 37 n + 10 }$

A) $\frac { n + 3 } { n - 7 }$
B) $\frac { n - 1 } { n + 2 }$
C) $\frac { 3 ( n + 1 ) } { 7 ( n - 2 ) }$
D) $\frac { 3 n + 1 } { 7 n - 2 }$

Question 5

Simplify the rational expression. $\frac { - 4 y ^ { 3 } } { 14 x y ^ { 2 } }$

A) $- \frac { 2 y } { 7 x }$
B) $- \frac { 2 x } { 7 y }$
C) $- \frac { 2 x y } { 7 }$
D) $\frac { 2 y ^ { 2 } } { 7 x }$

Question 6

Simplify the rational expression. $\frac { 4 x ^ { 2 } y + 8 x y ^ { 2 } - 12 y ^ { 3 } } { 30 x ^ { 3 } y - 20 x ^ { 2 } y ^ { 2 } - 10 x y ^ { 3 } }$

A) $\frac { 2 } { 5 x }$
B) $\frac { 2 ( x - 3 y ) } { 5 x y ( 3 x - y ) }$
C) $\frac { 2 ( x + 3 y ) } { 5 x ( 3 x + y ) }$
D) $\frac { 2 y ( x + 3 y ) } { 5 x ( x - 3 y ) }$

Question 7

Simplify the rational expression. $\frac { x y + c y + k x + c k } { x y + c y + q x + c q }$

A) $\frac { k - y } { q - y }$
B) x + c
C) $\frac { k } { q }$
D) $\frac { k + y } { q + y }$

Question 8

Simplify the rational expression. $\frac { 64 x ^ { 4 } + 27 x } { 8 x ^ { 3 } - 26 x ^ { 2 } - 24 x }$

A) $\frac { 16 x ^ { 2 } - 24 x + 9 } { 2 x ( x - 4 ) }$
B) $\frac { 16 x ^ { 2 } - 12 x + 9 } { 2 ( x - 4 ) }$
C) $\frac { 16 x ^ { 2 } + 12 x + 9 } { 2 ( x + 4 ) }$
D) $\frac { x \left( 16 x ^ { 2 } - 12 x + 9 \right) } { 2 ( x - 4 ) }$

Question 9

Express the rational number in reduced form. $\frac { 24 } { 30 }$

A) $\frac { 4 } { 5 }$
B) $\frac { 5 } { 6 }$
C) $\frac { 1 } { 5 }$
D) $\frac { 1 } { 2 }$

Question 10

Simplify the rational expression. $\frac { - 4 x ^ { 3 } y } { - 6 x y ^ { 4 } }$

A) $- \frac { 2 x ^ { 2 } } { 3 y ^ { 3 } }$
B) $\frac { 2 y } { 3 x }$
C) $\frac { 2 x ^ { 2 } } { 3 y ^ { 3 } }$
D) $- \frac { 2 x ^ { 3 } } { 3 y ^ { 4 } }$

Question 11

Simplify the rational expression. $\frac { a ^ { 2 } + 13 a + 40 } { a ^ { 2 } - 3 a - 40 }$

A) $\frac { a + 8 } { a - 5 }$
B) a + 5
C) $\frac { a + 8 } { a + 5 }$
D) $\frac { a + 8 } { a - 8 }$

Question 12

The rational express $\frac { x + 1 } { x - 4 }$ is meaninful for all values of x except $- 4$ .

Question 13

Simplify the rational expression. $\frac { x ^ { 4 } - 2 x ^ { 2 } - 48 } { 4 x ^ { 4 } + 29 x ^ { 2 } + 30 }$

A) $\frac { x ^ { 2 } - 8 } { 4 x ^ { 2 } + 5 }$
B) $\frac { x - 8 } { 4 x + 5 }$
C) $\frac { x + 8 } { 4 x + 5 }$
D) $\frac { x ^ { 2 } - 8 } { x ^ { 2 } + 5 }$

Question 14

A rational expression is the indicated quotient of two expressions.

Question 15

Express the rational number in reduced form. $\frac { 65 } { - 78 }$

A) $\frac { 1 } { 2 }$
B) $- \frac { 6 } { 7 }$
C) $\frac { 1 } { 6 }$
D) $- \frac { 5 } { 6 }$

Question 16

Simplify the rational expression. $\frac { 2 x ^ { 2 } + 4 x } { x ^ { 3 } + 8 }$

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

Question 17

Simplify the rational expression. $\frac { 5 y ^ { 2 } + 39 y + 28 } { 25 y ^ { 2 } - 16 }$

A) $\frac { y } { y - 4 }$
B) $\frac { y + 7 } { 5 y - 4 }$
C) $\frac { y - 7 } { y + 4 }$
D) $\frac { y + 7 } { 5 y - 16 }$

Question 18

The first step in simplifying a rational expression is to factor the numerator and the denominator, if possible.

Question 19

Simplify the rational expression. $\frac { x ^ { 2 } - 16 } { x ^ { 2 } + 4 x }$

A) $\frac { x + 4 } { x }$
B) $- \frac { 4 } { x }$
C) $\frac { x - 4 } { x }$
D) $\frac { 4 } { x }$

Question 20

Simplify the rational expression. $\frac { 44 x ^ { 3 } z } { - 48 x y z ^ { 2 } }$

A) $\frac { 11 x ^ { 2 } } { 12 z }$
B) $- \frac { 44 x } { 48 y z }$
C) $- \frac { 11 x ^ { 2 } } { 12 y z }$
D) $\frac { x ^ { 2 } } { 4 y z }$

Question 21

Simplify the rational expression. $\frac { 4 y - 4 x y } { x ^ { 2 } y - y }$

A) $- \frac { 4 } { x - 1 }$
B) $- \frac { 4 y } { x + y }$
C) $- \frac { 4 } { x + 1 }$
D) $- \frac { 4 y } { x + 1 }$
E) $- \frac { 4 } { x + y }$

Question 22

Simplify the rational expression.

Question 23

Simplify the rational expression.

Question 24

Simplify the rational expression. $\frac { n r - 12 - 4 n + 3 r } { n r + 15 + 3 r + 5 n }$

A) $\frac { r + 4 } { r - 5 }$
B) $- \frac { r + 4 } { r + 5 }$
C) $- \frac { r + 3 } { r + 4 }$
D) $\frac { r - 4 } { r + 5 }$

Question 25

Simplify the rational expression.

Question 26

Simplify the rational expression. $\frac { 4 x - 7 } { 7 - 4 x }$

A) 0
B) 1
C) 4 x + 7
D) 4 x $-$ 7
E) $-$ 1

Question 27

Simplify the rational expression.

Question 28

Simplify the rational expression.

Question 29

Simplify the rational expression. $\frac { a ^ { 2 } - 25 } { 5 - a }$

A) 5 $-$ a
B) a $-$ 4
C) $-$ a $-$ 4
D) a + 5
E) $-$ a $-$ 5

Question 30

Simplify the rational expression.

Question 31

Simplify the rational expression.

Question 32

Simplify the rational expression.

Question 33

Simplify the rational expression.

Question 34

Express the rational number in reduced form.

Question 35

Simplify the rational expression.

Question 36

Simplify the rational expression.

Question 37

Simplify the rational expression.

Question 38

Simplify the rational expression.

Question 39

Express the rational number in reduced form.

Question 40

Simplify the rational expression. $\frac { x ^ { 2 } + 2 x - 24 } { 20 - x - x ^ { 2 } }$

A) $\frac { x + 6 } { 4 - x }$
B) $\frac { x + 6 } { x - 4 }$
C) $\frac { x - 6 } { 5 - x }$
D) $- \frac { x + 6 } { x + 5 }$

Question 41

Perform the indicated operations involving rational numbers. Express the final answer in reduced form. $\frac { 4 } { 9 } \cdot \frac { 6 } { 17 } \div \frac { 1 } { 6 }$

A) $\frac { 1 } { 18 }$
B) $\frac { 17 } { 18 }$
C) $\frac { 16 } { 17 }$
D) $\frac { 1 } { 17 }$
E) $1$

Question 42

Perform the indicated operation involving rational expressions. $\frac { 5 x ^ { 8 } y ^ { 8 } } { 6 x ^ { 9 } y ^ { 7 } } \cdot \frac { 18 x ^ { 8 } } { 20 }$

A) $\frac { 3 x y ^ { 9 } } { 4 }$
B) $\frac { 4 x y ^ { 7 } } { 3 }$
C) $\frac { 3 x y ^ { 7 } } { 4 }$
D) $\frac { 4 x y ^ { 8 } } { 3 }$

Question 43

Simplify the rational expression.

Question 44

Simplify the rational expression.

Question 45

Perform the indicated operation involving rational expressions. $\frac { 7 x ^ { 4 } y ^ { 4 } } { 8 x ^ { 5 } y ^ { 3 } } \cdot \frac { 56 x ^ { 4 } } { 28 y } \cdot \frac { 4 x } { y ^ { 3 } }$

A) $\frac { 7 x ^ { 4 } } { y ^ { 3 } }$
B) $\frac { 4 x ^ { 4 } } { y ^ { 3 } }$
C) $\frac { 7 x ^ { 4 } } { y ^ { 4 } }$
D) $\frac { 7 x ^ { 3 } } { 4 y ^ { 3 } }$

Question 46

Perform the indicated operation involving rational expressions. $\frac { - 6 x y ^ { 4 } } { 7 y ^ { 2 } } \cdot \frac { 77 x ^ { 2 } y ^ { 3 } } { 24 y ^ { 2 } }$

A) $- \frac { 11 x ^ { 2 } y ^ { 2 } } { 4 }$
B) $- \frac { x ^ { 3 } y ^ { 3 } } { 11 }$
C) $- \frac { 11 x ^ { 3 } y ^ { 3 } } { 4 }$
D) $- \frac { 4 } { 11 x ^ { 3 } y ^ { 3 } }$

Question 47

Perform the indicated operations involving rational numbers. Express final answer in reduced form. $\frac { - 16 } { 8 } \div \frac { 64 } { 24 }$

A) $- \frac { 1 } { 2 }$
B) $- \frac { 4 } { 3 }$
C) $- \frac { 3 } { 4 }$
D) $- \frac { 3 } { 8 }$
E) $- \frac { 1 } { 8 }$

Question 48

In a division problem, change all division to multiplication in order to simplify.

Question 49

Perform the indicated operation involving rational numbers. $\frac { 3 } { 4 } \cdot \frac { 44 } { 39 }$

A) 13
B) 11
C) $\frac { 11 } { 13 }$
D) $\frac { 13 } { 11 }$

Question 50

Perform the indicated operation involving rational expressions. $\frac { 2 x + 4 } { 11 y } \cdot \frac { x ^ { 2 } + 5 } { x ^ { 2 } + 4 x + 4 }$

A) $\frac { x ^ { 2 } + 5 } { 11 y ( x + 2 ) }$
B) $\frac { 2 \left( x ^ { 2 } + 5 \right) } { y ( x + 2 ) }$
C) $\frac { 2 \left( x ^ { 2 } + 5 \right) } { 11 y ( x + 2 ) }$
D) $\frac { 2 ( x - 5 ) } { 11 y ( x - 2 ) }$

Question 51

Perform the indicated operation involving rational numbers. $\left( - \frac { 5 } { 6 } \right) \div \frac { 20 } { 42 }$

A) $-$ 4
B) $- \frac { 7 } { 4 }$
C) $- \frac { 4 } { 7 }$
D) $-$ 7

Question 52

Perform the indicated operation involving rational expressions. $\frac { 2 a ^ { 5 } b ^ { 5 } } { 3 a ^ { 4 } b ^ { 4 } } \cdot \frac { 9 a ^ { 6 } } { 8 a ^ { 4 } b ^ { 5 } }$

A) $\frac { 3 a ^ { 2 } } { 4 b ^ { 5 } }$
B) $\frac { 3 a ^ { 3 } } { 4 b ^ { 4 } }$
C) $\frac { 4 a ^ { 3 } } { 3 b ^ { 4 } }$
D) $\frac { 3 a ^ { 3 } } { b ^ { 5 } }$

Question 53

Simplify the rational expression.

Question 54

Simplify the rational expression. You will need to use factoring by grouping.

Question 55

Perform the indicated operation involving rational numbers. $\frac { - 6 } { 7 } \cdot \frac { 35 } { 78 }$

A) $- \frac { 5 } { 13 }$
B) $-$ 5
C) $- \frac { 13 } { 5 }$
D) $-$ 13

Question 56

The numbers $\frac { - 3 } { 7 }$ and $\frac { - 7 } { 3 }$ are multiplicative inverses.

Question 57

Simplify the rational expression. You will need to use factoring by grouping.

Question 58

To simplify a rational expression, divide out any common terms.

Question 59

The quotient of two rational expressions is the product obtained when multiplying the first expression by the reciprocal of the second.

Question 60

Perform the indicated operation involving rational numbers. $\frac { 2 } { - 3 } \cdot \frac { - 15 } { 16 }$

A) 5
B) $-$ 8
C) $- \frac { 8 } { 5 }$
D) $\frac { 5 } { 8 }$

Question 61

Perform the indicated operation involving rational expressions. Express final answer in simplest form.

Question 62

Perform the indicated operation involving rational expressions. $\frac { 2 x ^ { 9 } y ^ { 9 } } { 3 x ^ { 10 } y ^ { 8 } } \div \frac { 26 } { 9 x ^ { 9 } }$

A) $\frac { 3 x y ^ { 8 } } { 13 }$
B) $\frac { 13 x y ^ { 8 } } { 3 }$
C) $\frac { 13 x y ^ { 9 } } { 3 }$
D) $\frac { 3 x y ^ { 10 } } { 13 }$

Question 63

Perform the indicated operation involving rational expressions. Express final answer in simplest form.

Question 64

Perform the indicated operation involving rational expressions. $\frac { 7 t ^ { 5 } - 21 t ^ { 4 } } { t ^ { 3 } - 3 t ^ { 2 } } \cdot \frac { t ^ { 2 } - 6 t + 9 } { t ^ { 5 } - 9 t ^ { 3 } }$

A) $\frac { 7 t - 21 } { t ^ { 2 } + 3 t }$
B) $\frac { t - 3 } { t ^ { 2 } + 3 t }$
C) $\frac { 7 } { t ^ { 2 } + 3 t }$
D) $\frac { 7 t ^ { 2 } + 21 } { t ^ { 2 } + 3 t }$

Question 65

Perform the indicated operations involving rational expressions. Express the final answer in simplest form. $\frac { 4 n ^ { 2 } + 20 n - 24 } { 3 n ^ { 2 } + 10 n - 48 } \cdot \frac { 6 n ^ { 2 } - n - 40 } { 6 n ^ { 2 } + 9 n - 15 }$

A) $\frac { 3 } { 4 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 6 n ^ { 2 } - n - 40 } { 3 n ^ { 2 } + 10 n - 48 }$
D) $\frac { 4 } { 3 }$
E) $\frac { 4 } { 3 } \cdot \frac { 6 n ^ { 2 } - n - 40 } { 3 n ^ { 2 } + 10 n - 48 }$

Question 66

Perform the indicated operation involving rational expressions. $\frac { 2 x + 14 } { 11 y } \div \frac { x ^ { 2 } + 14 x + 49 } { x ^ { 2 } + 2 }$

A) $\frac { 2 x ^ { 2 } + 4 } { x y + 7 y }$
B) $\frac { x ^ { 2 } + 2 } { 11 x y + 77 y }$
C) $\frac { 2 x - 4 } { 11 x y - 77 y }$
D) $\frac { 2 x ^ { 2 } + 4 } { 11 x y + 77 y }$

Question 67

Perform the indicated operation involving rational expressions. $\frac { 3 a ^ { 5 } b ^ { 5 } } { 4 a ^ { 4 } c b ^ { 4 } } \div \frac { 6 c a ^ { 4 } b ^ { 5 } } { 20 a ^ { 6 } }$

A) $\frac { 5 a ^ { 2 } c ^ { 2 } } { 2 b ^ { 5 } }$
B) $\frac { 2 a ^ { 3 } } { 5 b ^ { 4 } }$
C) $\frac { 5 a ^ { 3 } } { 2 b ^ { 4 } c ^ { 2 } }$
D) $\frac { 5 a ^ { 3 } } { b ^ { 5 } c }$

Question 68

Perform the indicated operation involving rational expressions. $\frac { 16 n - 16 - 4 n ^ { 2 } } { 50 n - 125 - 5 n ^ { 2 } } \cdot \frac { 150 n - 375 - 15 n ^ { 2 } } { 144 - 96 n + 16 n ^ { 2 } }$

A) $- \frac { ( n - 2 ) ^ { 2 } } { ( n - 3 ) ^ { 2 } }$
B) $- \frac { 3 ( n - 2 ) ^ { 2 } } { 4 ( n - 3 ) ^ { 2 } }$
C) $\frac { n + 2 } { 4 ( n + 3 ) ^ { 2 } }$
D) $- \frac { ( n - 2 ) ^ { 2 } } { 4 ( n + 3 ) ^ { 2 } }$

Question 69

Perform the indicated operations involving rational expressions. Express the final answer in simplest form. $\frac { 3 x ^ { 2 } - 20 x + 25 } { 2 x ^ { 2 } - 7 x - 15 } \div \frac { 9 x ^ { 2 } - 3 x - 20 } { 12 x ^ { 2 } + 28 x + 15 }$

A) $\frac { 6 x - 5 } { 3 x - 2 }$
B) $\frac { 6 x + 5 } { 3 x + 4 }$
C) $\frac { 6 x + 9 } { 3 x + 4 }$
D) $\frac { 6 x - 5 } { 3 x - 4 }$
E) $\frac { 6 x + 9 } { 3 x + 2 }$

Question 70

Perform the indicated operation involving rational numbers. Express final answer in reduced form.

Question 71

Perform the indicated operation involving rational expressions. $\frac { n r + 8 n - 8 r - 64 } { n r + 4 n + 4 r + 16 } \cdot \frac { n ^ { 2 } - 16 } { n ^ { 3 } - 8 n ^ { 2 } }$

A) $\frac { n r - 4 r + 8 n + 32 } { m n + 4 n }$
B) $\frac { n r - 4 r + 8 n - 32 } { m ^ { 2 } + 4 n ^ { 2 } }$
C) $\frac { n r + 4 r + 8 n - 32 } { m ^ { 2 } - 4 n ^ { 2 } }$
D) $\frac { n r - 4 r + 8 n + 32 } { r ^ { 2 } + 4 n ^ { 2 } }$

Question 72

Perform the indicated operation involving rational numbers. Express final answer in reduced form.

Question 73

Perform the indicated operation involving rational expressions. $\frac { 8 x ^ { 8 } y ^ { 8 } } { 9 x ^ { 9 } y ^ { 7 } } \cdot \frac { 99 x ^ { 8 } } { 32 y } \div \frac { y ^ { 7 } } { 4 x }$

A) $\frac { 11 x ^ { 8 } } { y ^ { 8 } }$
B) $\frac { 11 x ^ { 8 } } { y ^ { 7 } }$
C) $\frac { 4 x ^ { 8 } } { y ^ { 7 } }$
D) $\frac { 11 x ^ { 7 } } { 4 y ^ { 7 } }$

Question 74

Perform the indicated operation involving rational expressions. Express final answer in simplest form.

Question 75

Perform the indicated operations involving rational numbers. Express final answer in reduced form.

Question 76

Perform the indicated operations involving rational numbers. Express the final answer in reduced form.

Question 77

Perform the indicated operation involving rational numbers. Express final answer in reduced form.

Question 78

Perform the indicated operation involving rational expressions. $\frac { 8 a ^ { 2 } - 64 a } { a ^ { 3 } - 8 a ^ { 2 } } \cdot \frac { a ^ { 2 } - 16 a + 64 } { a ^ { 2 } - 64 }$

A) $\frac { 8 a ^ { 2 } + 64 } { a ^ { 2 } + 8 a }$
B) $\frac { a - 8 } { a ^ { 2 } + 8 a }$
C) $\frac { 8 } { a ^ { 2 } + 8 a }$
D) $\frac { 8 a - 64 } { a ^ { 2 } + 8 a }$

Question 79

Perform the indicated operation involving rational expressions. $\frac { 3 a ^ { 2 } - 18 a } { a ^ { 3 } - 7 a ^ { 2 } } \cdot \frac { a ^ { 2 } - 14 a + 49 } { a ^ { 2 } - 36 }$

A) $\frac { 3 a - 21 } { a ^ { 2 } + 6 a }$
B) $\frac { 21 } { a ^ { 2 } + 3 a }$
C) $\frac { 3 a ^ { 2 } + 21 } { a ^ { 2 } + 6 a }$
D) $\frac { 3 } { a ^ { 2 } + 6 a }$
E) $\frac { a - 7 } { a ^ { 2 } + 6 a }$

Question 80

Perform the indicated operations involving rational expressions. Express the final answer in simplest form. $\frac { 12 x ^ { 2 } - 5 x - 3 } { 8 x ^ { 2 } - 2 x - 3 } \div \frac { 21 x ^ { 2 } + x - 2 } { 2 x ^ { 2 } - 17 x - 9 }$

A) $\frac { x - 8 } { 7 x - 3 }$
B) $\frac { x - 9 } { 7 x - 2 }$
C) $\frac { x + 9 } { 7 x + 2 }$
D) $\frac { x + 9 } { 7 x + 3 }$
E) $\frac { x - 8 } { 7 x - 2 }$