Lesson 28: Simplifying Rational Expressions

New Concept

A rational expression is simplified when there are no common factors, other than 1, in the numerator and denominator.

What’s next

Next, you'll learn the rules for simplifying these expressions, starting with the Quotient of Powers Property and factoring.

For $a \neq 0$, and integers $m$ and $n$,

Excluded value is 0. Simplify:

Imagine a power showdown! When dividing powers with the same base, just subtract the bottom exponent from the top one. It's a shortcut to clean up fractions with monomial terms, making big scary expressions like $\frac{x^9}{x^2}$ much simpler. Remember to find any excluded values first, where the denominator would be zero, before you start simplifying!

A rational expression is simplified when there are no common factors, other than 1, in the numerator and denominator. This is achieved by dividing out common factors.

Excluded value is -3. Simplify:

This is like a matching game for fractions! First, find any 'excluded values' that make the denominator zero. Then, factor the top and bottom expressions completely. If you spot the exact same factor on both levels, you can cancel them out. It’s the ultimate clean-up move for messy polynomial fractions, revealing a much simpler expression underneath.

To simplify expressions with opposite binomials, such as $\frac{a-b}{b-a}$, factor out a $-1$ from one of the binomials. For example, $(b-a)$ can be rewritten as $-1(a-b)$.

Excluded value is 4. Simplify:

Ever see factors that are almost identical but backward, like $(x-5)$ and $(5-x)$? Don't worry! Use a clever trick by factoring out a $-1$ from one of them. This flips its signs, making it a perfect match for the other factor. Now you can cancel them out, leaving just a $-1$ behind. It's a ninja move for simplifying!