# 8.5 Simplify complex rational expressions

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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: $\frac{\frac{3}{5}}{\frac{9}{10}}.$
If you missed this problem, review [link] .
2. Simplify: $\frac{1-\frac{1}{3}}{{4}^{2}+4·5}.$
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:

$\frac{\frac{3}{4}}{\frac{5}{8}}\phantom{\rule{4em}{0ex}}\frac{\frac{x}{2}}{\frac{xy}{6}}$

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:

$\frac{\frac{4}{y-3}}{\frac{8}{{y}^{2}-9}}\phantom{\rule{7em}{0ex}}\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}\phantom{\rule{7em}{0ex}}\frac{\frac{2}{x+6}}{\frac{4}{x-6}-\frac{4}{{x}^{2}-36}}$

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.

$\frac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-8x+3}{{x}^{2}-5x+6}}$

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

$\left(\frac{6{x}^{2}-7x+2}{4x-8}\right)÷\left(\frac{2{x}^{2}-8x+3}{{x}^{2}-5x+6}\right)$

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: $\frac{\frac{4}{y-3}}{\frac{8}{{y}^{2}-9}}.$

## Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{\frac{4}{y-3}}{\frac{8}{{y}^{2}-9}}\hfill \\ \\ \\ \text{Rewrite the complex fraction as division.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4}{y-3}÷\frac{8}{{y}^{2}-9}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite as the product of first times the}\hfill \\ \text{reciprocal of the second.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4}{y-3}·\frac{{y}^{2}-9}{8}\hfill \\ \\ \\ \text{Multiply.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4\left({y}^{2}-9\right)}{8\left(y-3\right)}\hfill \\ \\ \\ \text{Factor to look for common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4\left(y-3\right)\left(y+3\right)}{4·2\left(y-3\right)}\hfill \\ \\ \\ \text{Remove common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\overline{)4}\overline{)\left(y-3\right)}\left(y+3\right)}{\overline{)4}·2\overline{)\left(y-3\right)}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{y+3}{2}\hfill \end{array}$

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 $y=-3$ .

Simplify: $\frac{\frac{2}{{x}^{2}-1}}{\frac{3}{x+1}}.$

$\frac{2}{3\left(x-1\right)}$

Simplify: $\frac{\frac{1}{{x}^{2}-7x+12}}{\frac{2}{x-4}}.$

$\frac{1}{2\left(x-3\right)}$

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: $\frac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}-\frac{1}{3}}.$

## 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.

Simplify: $\frac{\frac{1}{2}+\frac{2}{3}}{\frac{5}{6}+\frac{1}{12}}.$

$\frac{14}{11}$

Simplify: $\frac{\frac{3}{4}-\frac{1}{3}}{\frac{1}{8}+\frac{5}{6}}.$

$\frac{10}{23}$

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

Simplify: $\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}.$

## Solution

Simplify: $\frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x}-\frac{1}{y}}.$

$\frac{y+x}{y-x}$

Simplify: $\frac{\frac{1}{a}+\frac{1}{b}}{\frac{1}{{a}^{2}}-\frac{1}{{b}^{2}}}.$

$\frac{ab}{b-a}$

## 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.

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?
Uno de los ángulos suplementario es 4° más que 1/3 del otro ángulo encuentra las medidas de cada uno de los angulos
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?
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?
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.
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?
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?
whats the coefficient of 17x
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