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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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Questions & Answers

sin^4+sin^2=1, prove that tan^2-tan^4+1=0
SAYANTANI Reply
what is the formula used for this question? "Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
Kuz Reply
i don't need help solving it I just need a memory jogger please.
Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
Mintah Reply
its a very simple
Kavita
gave your expression then i solve
Kavita
Hy guys, I have a problem when it comes on solving equations and expressions, can you help me 😭😭
Thuli
Tomorrow its an revision on factorising and Simplifying...
Thuli
ok sent the quiz
kurash
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Kavita
Hi
Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
Masum
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Kavita
No sister.
Masum
no I m right
Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
Joju Reply
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
musah Reply
what is sine?
tae Reply
what is the standard form of 1
Sanjana Reply
1×10^0
Akugry
Evalute exponential functions
Sujata Reply
30
Shani
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Jaya Reply
Will be with you shortly
Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
answer is 2, 3, 4
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
Ashutosh Reply
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
Akugry
hi
Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
Practice Key Terms 2

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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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