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Given a complex rational expression, simplify it.

  1. Combine the expressions in the numerator into a single rational expression by adding or subtracting.
  2. Combine the expressions in the denominator into a single rational expression by adding or subtracting.
  3. Rewrite as the numerator divided by the denominator.
  4. Rewrite as multiplication.
  5. Multiply.
  6. Simplify.

Simplifying complex rational expressions

Simplify: y + 1 x x y .

Begin by combining the expressions in the numerator into one expression.

y x x + 1 x    Multiply by  x x  to get LCD as denominator . x y x + 1 x x y + 1 x    Add numerators .

Now the numerator is a single rational expression and the denominator is a single rational expression.

x y + 1 x x y

We can rewrite this as division, and then multiplication.

x y + 1 x ÷ x y x y + 1 x y x Rewrite as multiplication . y ( x y + 1 ) x 2 Multiply .
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Simplify: x y y x y

x 2 y 2 x y 2

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Can a complex rational expression always be simplified?

Yes. We can always rewrite a complex rational expression as a simplified rational expression.

Access these online resources for additional instruction and practice with rational expressions.

Key concepts

  • Rational expressions can be simplified by cancelling common factors in the numerator and denominator. See [link] .
  • We can multiply rational expressions by multiplying the numerators and multiplying the denominators. See [link] .
  • To divide rational expressions, multiply by the reciprocal of the second expression. See [link] .
  • Adding or subtracting rational expressions requires finding a common denominator. See [link] and [link] .
  • Complex rational expressions have fractions in the numerator or the denominator. These expressions can be simplified. See [link] .

Section exercises

Verbal

How can you use factoring to simplify rational expressions?

You can factor the numerator and denominator to see if any of the terms can cancel one another out.

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How do you use the LCD to combine two rational expressions?

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Tell whether the following statement is true or false and explain why: You only need to find the LCD when adding or subtracting rational expressions.

True. Multiplication and division do not require finding the LCD because the denominators can be combined through those operations, whereas addition and subtraction require like terms.

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Algebraic

For the following exercises, simplify the rational expressions.

x 2 16 x 2 5 x + 4

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y 2 + 10 y + 25 y 2 + 11 y + 30

y + 5 y + 6

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6 a 2 24 a + 24 6 a 2 24

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9 b 2 + 18 b + 9 3 b + 3

3 b + 3

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2 x 2 + 7 x 4 4 x 2 + 2 x 2

x + 4 2 x + 2

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6 x 2 + 5 x 4 3 x 2 + 19 x + 20

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a 2 + 9 a + 18 a 2 + 3 a 18

a + 3 a 3

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3 c 2 + 25 c 18 3 c 2 23 c + 14

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12 n 2 29 n 8 28 n 2 5 n 3

3 n 8 7 n 3

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For the following exercises, multiply the rational expressions and express the product in simplest form.

x 2 x 6 2 x 2 + x 6 2 x 2 + 7 x 15 x 2 9

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c 2 + 2 c 24 c 2 + 12 c + 36 c 2 10 c + 24 c 2 8 c + 16

c 6 c + 6

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2 d 2 + 9 d 35 d 2 + 10 d + 21 3 d 2 + 2 d 21 3 d 2 + 14 d 49

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10 h 2 9 h 9 2 h 2 19 h + 24 h 2 16 h + 64 5 h 2 37 h 24

1

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6 b 2 + 13 b + 6 4 b 2 9 6 b 2 + 31 b 30 18 b 2 3 b 10

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2 d 2 + 15 d + 25 4 d 2 25 2 d 2 15 d + 25 25 d 2 1

d 2 25 25 d 2 1

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6 x 2 5 x 50 15 x 2 44 x 20 20 x 2 7 x 6 2 x 2 + 9 x + 10

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t 2 1 t 2 + 4 t + 3 t 2 + 2 t 15 t 2 4 t + 3

t + 5 t + 3

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2 n 2 n 15 6 n 2 + 13 n 5 12 n 2 13 n + 3 4 n 2 15 n + 9

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36 x 2 25 6 x 2 + 65 x + 50 3 x 2 + 32 x + 20 18 x 2 + 27 x + 10

6 x 5 6 x + 5

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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