# Simplifying Rational Expressions

In this video, we are going to simplify rational expressions.

For example:
$\frac{3x^4}{12x^5}$
It’s just like reducing fractions. Begin by simplifying the constants.
$\frac{3}{12}$ would be $\frac{1}{4}$
Let’s now look at the exponents. The numerator is to the fourth power while the denominator is to the fifth power. As a result, there will be one x in the denominator but not in the numerator.
$\frac{1x^4}{4x^5}$
$\frac{1}{4x}$

Now let’s look at an example where there is an operation taking place in the numerator.
$\frac{24x^5-15x^3}{6x^4}$
Here we can separate into two separate fractions
$\frac{24x^5}{6x^4}$ and $\frac{-15x^3}{6x^4}$
Then we can simplify each one and get a final answer of
$6x-\frac{-5}{2x}$

When there is an operation in the denominator, we cannot simply do the same thing. Instead we must factor each term by its greatest common factor. For example:
$\frac{6x^4}{24x^5-15x^3}$
The greatest common factor for all three terms is $3x^3$. So if we take out a $3x^3$ from each term, we are left with
$\frac{3x^3(2x)}{3x^3(8x^2-5)}$
Then, the $3x^3$ in the numerator and denominator will cancel out, leaving us with a final answer of
$\frac{2x}{8x^2-5}$

Another type of rational expression may look like
$\frac{x^2-x-12}{x^2-2x-8}$
To solve something like this, we can factor the quadratic in the numerator and denominator. In this example, we would have
$\frac{(x-4)(x+3)}{(x-4)(x+2)}$
From here, we could cancel out the (x-4) from the numerator and denominator, leaving us with
$\frac{x+3}{x+2}$