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w 2 3 w 28 49 w 2

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Multiply and Divide Rational Expressions

Multiply Rational Expressions

In the following exercises, multiply.

2 x y 2 8 y 3 · 16 y 24 x

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3 a 2 + 21 a a 2 + 6 a 7 · a 1 a b

3 b

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5 z 2 5 z 2 + 40 z + 35 · z 2 1 3 z

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Divide Rational Expressions

In the following exercises, divide.

t 2 4 t + 12 t 2 + 8 t + 12 ÷ t 2 36 6 t

6 t ( t + 6 ) ( t + 6 )

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r 2 16 4 ÷ r 3 64 2 r 2 8 r + 32

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11 + w w 9 ÷ 121 w 2 9 w

1 11 + w

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3 y 2 12 y 63 4 y + 3 ÷ ( 6 y 2 42 y )

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c 2 64 3 c 2 + 26 c + 16 c 2 4 c 32 15 c + 10

5 c + 4

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8 m 2 8 m m 4 · m 2 + 2 m 24 m 2 + 7 m + 10 ÷ 2 m 2 6 m m + 5

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Add and Subtract Rational Expressions with a Common Denominator

Add Rational Expressions with a Common Denominator

In the following exercises, add.

4 a 2 2 a 1 1 2 a 1

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p 2 + 10 p p + 5 + 25 p + 5

p + 5

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Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

d 2 d + 4 3 d + 28 d + 4

d + 7

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z 2 z + 10 100 z + 10

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4 q 2 q + 3 q 2 + 6 q + 5 3 q 2 q 6 q 2 + 6 q + 5

q 3 q + 5

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5 t + 4 t + 3 t 2 25 4 t 2 8 t 32 t 2 25

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Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add and subtract.

18 w 6 w 1 + 3 w 2 1 6 w

15 w + 2 6 w 1

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a 2 + 3 a a 2 4 3 a 8 4 a 2

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2 b 2 + 3 b 15 b 2 49 b 2 + 16 b 1 49 b 2

3 b 2 b + 7

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8 y 2 10 y + 7 2 y 5 + 2 y 2 + 7 y + 2 5 2 y

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Add and Subtract Rational Expressions With Unlike Denominators

Find the Least Common Denominator of Rational Expressions

In the following exercises, find the LCD.

4 m 2 3 m 10 , 2 m m 2 m 20

( m + 2 ) ( m 5 ) ( m + 4 )

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6 n 2 4 , 2 n n 2 4 n + 4

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5 3 p 2 + 17 p 6 , 2 m 3 p 2 23 p 8

( 3 p + 1 ) ( p + 6 ) ( p + 8 )

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Find Equivalent Rational Expressions

In the following exercises, rewrite as equivalent rational expressions with the given denominator.

Rewrite as equivalent rational expressions with denominator ( m + 2 ) ( m 5 ) ( m + 4 ) :

4 m 2 3 m 10 , 2 m m 2 m 20 .
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Rewrite as equivalent rational expressions with denominator ( n 2 ) ( n 2 ) ( n + 2 ) :

6 n 2 4 n + 4 , 2 n n 2 4 .

6 n + 12 ( n 2 ) ( n 2 ) ( n + 2 ) ,
2 n 2 4 n ( n 2 ) ( n 2 ) ( n + 2 )

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Rewrite as equivalent rational expressions with denominator ( 3 p + 1 ) ( p + 6 ) ( p + 8 ) :

5 3 p 2 + 19 p + 6 , 7 p 3 p 2 + 25 p + 8
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Add Rational Expressions with Different Denominators

In the following exercises, add.

2 c 2 + 9 c + 3

11 c 12 ( c 2 ) ( c + 3 )

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3 d d 2 9 + 5 d 2 + 6 d + 9

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2 x x 2 + 10 x + 24 + 3 x x 2 + 8 x + 16

5 x 2 + 26 x ( x + 4 ) ( x + 4 ) ( x + 6 )

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5 q p 2 q p 2 + 4 q q 2 1

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Subtract Rational Expressions with Different Denominators

In the following exercises, subtract and add.

3 v v + 2 v + 2 v + 8

2 ( v 2 + 10 v 2 ) ( v + 2 ) ( v + 8 )

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−3 w 15 w 2 + w 20 w + 2 4 w

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7 m + 3 m + 2 5

2 m 7 m + 2

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n n + 3 + 2 n 3 n 9 n 2 9

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8 d d 2 64 4 d + 8

4 d 8

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Simplify Complex Rational Expressions

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify.

5 a a + 2 10 a 2 a 2 4

a 2 2 a

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x 3 x x + 5 1 x + 5 + 1 x 5

( x 8 ) ( x 5 ) 2

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Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

6 + 2 q 4 5 q + 4

( q 2 ) ( q + 4 ) 5 ( q 4 )

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3 a 2 1 b 1 a + 1 b 2

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2 z 2 49 + 1 z + 7 9 z + 7 + 12 z 7

z 5 23 z + 21

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3 y 2 4 y 32 2 y 8 + 1 y + 4

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Solve Rational Equations

Solve Rational Equations

In the following exercises, solve.

1 b 2 + 1 b + 2 = 3 b 2 4

3 2

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3 q + 8 2 q 2 = 1

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v 15 v 2 9 v + 18 = 4 v 3 + 2 v 6

no solution

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Solve a Rational Equation for a Specific Variable

In the following exercises, solve for the indicated variable.

V l = h w for l

l = V h w

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x = y + 5 z 7 for z

z = y + 5 + 7 x x

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Solve Proportion and Similar Figure Applications Similarity

Solve Proportions

In the following exercises, solve.

In the following exercises, solve using proportions.

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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