# 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.

He charges $125 per job. His monthly expenses are$1,600. How many jobs must he work in order to make a profit of at least $2,400? Alicia Reply at least 20 Ayla what are the steps? Alicia 6.4 jobs Grahame 32 Grahame what is algebra Azhar Reply repeated addition and subtraction of the order of operations. i love algebra I'm obsessed. Shemiah hi Krekar One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag? Leanna Reply rectangular field solutions Navin Reply What is this? Donna the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is anas Reply ? Choli a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190 Usman Stella bought a dinette set on sale for$725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount? Manhwa Reply 44.19% Scott 40.22% Terence 44.2% Orlando I don't know Donna if you want the discounted price subtract$725 from $1299. then divide the answer by$1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2%
Orlando
you could also just divide $725/$1299 and then subtract it from 1. then you get the same answer.
Orlando
p mulripied-5 and add 30 to it
Tausif
Tausif
Can you explain further
p mulripied-5 and add to 30
Tausif
-5p+30?
Corey
p=-5+30
Jacob
How do you find divisible numbers without a calculator?
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was$1,585. What is the amount of each loan?
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?
66miles/hour
snigdha
How did you work it out?
Esther
s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr
Orlando
hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
what do you need help in?
Felix
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
think about it like you lost $19 (-19), then found$7(+7). Totally you lost just $12 (-12) Annushka I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-\$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
I don't see where the answers are.
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