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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

what are you up to?
Mark Reply
nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
jai
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jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
cos(- z)=cos z
Mustafa
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins 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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