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By the end of this section, you will be able to:
  • Simplify a complex rational expression by writing it as division
  • Simplify a complex rational expression by using the LCD

Before you get started, take this readiness quiz.

If you miss a problem, go back to the section listed and review the material.

  1. Simplify: 3 5 9 10 .
    If you missed this problem, review [link] .
  2. Simplify: 1 1 3 4 2 + 4 · 5 .
    If you missed this problem, review [link] .

Complex fractions are fractions in which the numerator or denominator contains a fraction. In Chapter 1 we simplified complex fractions like these:

3 4 5 8 x 2 x y 6

In this section we will simplify complex rational expressions , which are rational expressions with rational expressions in the numerator or denominator.

Complex rational expression

A complex rational expression    is a rational expression in which the numerator or denominator contains a rational expression.

Here are a few complex rational expressions:

4 y 3 8 y 2 9 1 x + 1 y x y y x 2 x + 6 4 x 6 4 x 2 36

Remember, we always exclude values that would make any denominator zero.

We will use two methods to simplify complex rational expressions.

Simplify a complex rational expression by writing it as division

We have already seen this complex rational expression earlier in this chapter.

6 x 2 7 x + 2 4 x 8 2 x 2 8 x + 3 x 2 5 x + 6

We noted that fraction bars tell us to divide, so rewrote it as the division problem

( 6 x 2 7 x + 2 4 x 8 ) ÷ ( 2 x 2 8 x + 3 x 2 5 x + 6 )

Then we multiplied the first rational expression by the reciprocal of the second, just like we do when we divide two fractions.

This is one method to simplify rational expressions. We write it as if we were dividing two fractions.

Simplify: 4 y 3 8 y 2 9 .

Solution

4 y 3 8 y 2 9 Rewrite the complex fraction as division. 4 y 3 ÷ 8 y 2 9 Rewrite as the product of first times the reciprocal of the second. 4 y 3 · y 2 9 8 Multiply. 4 ( y 2 9 ) 8 ( y 3 ) Factor to look for common factors. 4 ( y 3 ) ( y + 3 ) 4 · 2 ( y 3 ) Remove common factors. 4 ( y 3 ) ( y + 3 ) 4 · 2 ( y 3 ) Simplify. y + 3 2

Are there any value(s) of y that should not be allowed? The simplified rational expression has just a constant in the denominator. But the original complex rational expression    had denominators of y 3 and y 2 9 . This expression would be undefined if y = 3 or y = −3 .

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Simplify: 2 x 2 1 3 x + 1 .

2 3 ( x 1 )

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Simplify: 1 x 2 7 x + 12 2 x 4 .

1 2 ( x 3 )

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Fraction bars act as grouping symbols. So to follow the Order of Operations, we simplify the numerator and denominator as much as possible before we can do the division.

Simplify: 1 3 + 1 6 1 2 1 3 .

Solution

.
Simplify the numerator and denominator.
Find the LCD and add the fractions in the numerator.
Find the LCD and add the fractions in the denominator.
.
Simplify the numerator and denominator. .
Simplify the numerator and denominator, again. .
Rewrite the complex rational expression as a division problem. .
Multiply the first times by the reciprocal of the second. .
Simplify. .

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Simplify: 1 2 + 2 3 5 6 + 1 12 .

14 11

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Simplify: 3 4 1 3 1 8 + 5 6 .

10 23

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How to simplify a complex rational expression by writing it as division

Simplify: 1 x + 1 y x y y x .

Solution

The above image has three columns. The image shows steps on how to divide complex rational expressions in three steps. Step one is to simplify the numerator and denominator. We will simplify the sum in the numerator and difference in the denominator for the example 1 divided by x plus 1 divided by y divided by x divided by y minus y divided by x. Find a common denominator and add the fractions in the numerator and find a common denominator and subtract the fractions in the numerator to get 1 times y divided by x times y plus 1 times x divided by y times x divided by x times x divided by y times x minus y times y divided by x times y. Then, we get y divided by x y plus x plus x y divided by x squared divided by x y minus y squared divided by x y. We now have just one rational expression in the numerator and one in the denominator, y plus x divided by x y divided by x squared minus y squared divided by x y. Step two is to rewrite the complex rational expression as a division problem. We write the numerator divided by the denominator. Step three is to divide the expressions. Multiply the first by the reciprocal of the second to get y plus x divided by x y times x y divided by x squared minus y squared. Factor any expressions if possible. We now have x y times y plus x divided by x y times x minus y times x plus y. Remove common factors. Cross out x, y and y plus x from the numerator. Cross out x, y and x plus y from the denominator. Simplify to get 1 divided by x minus y.
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Simplify: 1 x + 1 y 1 x 1 y .

y + x y x

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Simplify: 1 a + 1 b 1 a 2 1 b 2 .

a b b a

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Simplify a complex rational expression by writing it as division.

  1. Simplify the numerator and denominator.
  2. Rewrite the complex rational expression as a division problem.
  3. Divide the expressions.

Questions & Answers

what is the answer for this 3×9+28÷4-8
What Reply
315
lashonna
how do you do xsquard+7x+10=0
What
(x + 2)(x + 5), then set each factor to zero and solve for x. so, x = -2 and x = -5.
bruce
I skipped it
What
In 10 years, the population of Detroit fell from 950,000 to about 712,500. Find the percent decrease.
Jenise Reply
how do i set this up
Jenise
25%
Melissa
25 percent
Muzamil
950,000 - 712,500 = 237,500. 237,500 / 950,000 = .25 = 25%
Melissa
I've tried several times it won't let me post the breakdown of how you get 25%.
Melissa
Subtract one from the other to get the difference. Then take that difference and divided by 950000 and you will get .25 aka 25%
Melissa
Finally 👍
Melissa
one way is to set as ratio: 100%/950000 = x% / 712500, which yields that 712500 is 75% of the initial 950000. therefore, the decrease is 25%.
bruce
twenty five percent...
Jeorge
thanks melissa
Jeorge
950000-713500 *100 and then divide by 950000 = 25
Muzamil
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
jamie Reply
6t+3
Melissa
6t +3
Bollywood
Tricia got a 6% raise on her weekly salary. The raise was $30 per week. What was her original salary?
Iris Reply
let us suppose her original salary is 'm'. so, according to the given condition, m*(6/100)=30 m= (30*100)/6 m= 500 hence, her original salary is $500.
Simply
28.50
Toi
thanks
Jeorge
How many pounds of nuts selling for $6 per pound and raisins selling for $3 per pound should Kurt combine to obtain 120 pounds of trail mix that cost him $5 per pound?
Valeria Reply
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that code $20 per square foot. How many square feet of each tile should she use so that the overal cost of he backsplash will be $10 per square foot?
Nia Reply
I need help with maths can someone help me plz.. is there a wats app group?
Cindy Reply
WY need
Fernando
How did you get $750?
Laura Reply
if y= 2x+sinx what is dy÷dx
formon25 Reply
does it teach you how to do algebra if you don't know how
Kate Reply
Liam borrowed a total of $35,000 to pay for college. He pays his parents 3% interest on the $8,000 he borrowed from them and pays the bank 6.8% on the rest. What average interest rate does he pay on the total $35,000? (Round your answer to the nearest tenth of a percent.)
Lara Reply
exact definition of length by bilbao
francemichael Reply
the definition of length
francemichael Reply
literal meaning of length
francemichael
exact meaning of length
francemichael
exact meaning of length
francemichael
how many typos can we find...?
Chris Reply
5
Joseph
In the LCM Prime Factors exercises, the LCM of 28 and 40 is 280. Not 420!
Chris Reply
Practice Key Terms 1

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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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