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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}

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