# 1.6 Rational expressions  (Page 3/6)

 Page 3 / 6

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

Begin by combining the expressions in the numerator into one expression.

Now the numerator is a single rational expression and the denominator is a single rational expression.

$\frac{\frac{xy+1}{x}}{\frac{x}{y}}$

We can rewrite this as division, and then multiplication.

Simplify: $\frac{\frac{x}{y}-\frac{y}{x}}{y}$

$\frac{{x}^{2}-{y}^{2}}{x{y}^{2}}$

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] .
• Complex rational expressions have fractions in the numerator or the denominator. These expressions can be simplified. See [link] .

## 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.

How do you use the LCD to combine two rational expressions?

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.

## Algebraic

For the following exercises, simplify the rational expressions.

$\frac{{x}^{2}-16}{{x}^{2}-5x+4}$

$\frac{{y}^{2}+10y+25}{{y}^{2}+11y+30}$

$\frac{y+5}{y+6}$

$\frac{6{a}^{2}-24a+24}{6{a}^{2}-24}$

$\frac{9{b}^{2}+18b+9}{3b+3}$

$3b+3$

$\frac{m-12}{{m}^{2}-144}$

$\frac{2{x}^{2}+7x-4}{4{x}^{2}+2x-2}$

$\frac{x+4}{2x+2}$

$\frac{6{x}^{2}+5x-4}{3{x}^{2}+19x+20}$

$\frac{{a}^{2}+9a+18}{{a}^{2}+3a-18}$

$\frac{a+3}{a-3}$

$\frac{3{c}^{2}+25c-18}{3{c}^{2}-23c+14}$

$\frac{12{n}^{2}-29n-8}{28{n}^{2}-5n-3}$

$\frac{3n-8}{7n-3}$

For the following exercises, multiply the rational expressions and express the product in simplest form.

$\frac{{x}^{2}-x-6}{2{x}^{2}+x-6}\cdot \frac{2{x}^{2}+7x-15}{{x}^{2}-9}$

$\frac{{c}^{2}+2c-24}{{c}^{2}+12c+36}\cdot \frac{{c}^{2}-10c+24}{{c}^{2}-8c+16}$

$\frac{c-6}{c+6}$

$\frac{2{d}^{2}+9d-35}{{d}^{2}+10d+21}\cdot \frac{3{d}^{2}+2d-21}{3{d}^{2}+14d-49}$

$\frac{10{h}^{2}-9h-9}{2{h}^{2}-19h+24}\cdot \frac{{h}^{2}-16h+64}{5{h}^{2}-37h-24}$

$1$

$\frac{6{b}^{2}+13b+6}{4{b}^{2}-9}\cdot \frac{6{b}^{2}+31b-30}{18{b}^{2}-3b-10}$

$\frac{2{d}^{2}+15d+25}{4{d}^{2}-25}\cdot \frac{2{d}^{2}-15d+25}{25{d}^{2}-1}$

$\frac{{d}^{2}-25}{25{d}^{2}-1}$

$\frac{6{x}^{2}-5x-50}{15{x}^{2}-44x-20}\cdot \frac{20{x}^{2}-7x-6}{2{x}^{2}+9x+10}$

$\frac{{t}^{2}-1}{{t}^{2}+4t+3}\cdot \frac{{t}^{2}+2t-15}{{t}^{2}-4t+3}$

$\frac{t+5}{t+3}$

$\frac{2{n}^{2}-n-15}{6{n}^{2}+13n-5}\cdot \frac{12{n}^{2}-13n+3}{4{n}^{2}-15n+9}$

$\frac{36{x}^{2}-25}{6{x}^{2}+65x+50}\cdot \frac{3{x}^{2}+32x+20}{18{x}^{2}+27x+10}$

$\frac{6x-5}{6x+5}$

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar