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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

Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be?
74.25 × .16 then get the total and that will be your tip
David
$74.25 x 0.16 = $11.88 total bill: $74.25 + $11.88 = $86.13
ericka
yes and tip 16% will be $11.88
David
what is the shorter way to do it
Cesar Reply
Priam has dimes and pennies in a cup holder in his car. The total value of the coins is $4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?
Alexandra Reply
Uno de los ángulos suplementario es 4° más que 1/3 del otro ángulo encuentra las medidas de cada uno de los angulos
Enith Reply
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Alexandra Reply
I hope this is correct, x=cooler 1 5x=cooler 2 x + 5x = 48 6x=48 ×=8 gallons 5×=40 gallons
ericka
Priam has pennies and dimes in a cup holder in his car. The total value of the coins is $4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup?
Cecilia Reply
Arnold invested $64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received $4500 in interest in one year
Heidi Reply
List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.
Elbert Reply
Avery and Caden have saved $27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
Leika Reply
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine
Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his rectangular front yard. The length is five feet less than five times the width. Find the length and width of the fencing.
Ericka Reply
Mario invested $475 in $45 and $25 stock shares. The number of $25 shares was five less than three times the number of $45 shares. How many of each type of share did he buy?
Jawad Reply
let # of $25 shares be (x) and # of $45 shares be (y) we start with $25x + $45y=475, right? we are told the number of $25 shares is 3y-5) so plug in this for x. $25(3y-5)+$45y=$475 75y-125+45y=475 75y+45y=600 120y=600 y=5 so if #$25 shares is (3y-5) plug in y.
Joshua
will every polynomial have finite number of multiples?
cricket Reply
a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910.
David Reply
. A cashier has 54 bills, all of which are $10 or $20 bills. The total value of the money is $910. How many of each type of bill does the cashier have?
jojo Reply
whats the coefficient of 17x
Dwayne Reply
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne
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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