Lesson 51: Simplifying Rational Expressions with Like Denominators

New Concept

All values that would make the denominator equal zero are excluded.

What’s next

Next, you'll practice identifying these excluded values and use factoring to simplify rational expressions to their most basic forms.

Property

To find the excluded values of a rational expression, set the denominator equal to zero and solve for the variable. Any value that makes the denominator zero is excluded.

Examples

For the expression $\frac{7}{3x}$, set $3x=0$. The excluded value is $x \neq 0$.
For the expression $\frac{k-2}{k+5}$, set $k+5=0$. The excluded value is $k \neq -5$.
For the expression $\frac{m}{2m-8}$, set $2m-8=0$. The excluded value is $m \neq 4$.

Explanation

Think of the denominator as the ground floor of a building; it can't be zero, or everything collapses! We find the numbers that would cause this math disaster and ban them from the expression. It’s like being a bouncer for your fraction, keeping out the troublemaking zeros.

Property

To simplify a rational expression, factor the numerator and denominator. Then, divide out any common factors that appear in both.

Examples

$\frac{10x^3}{15x} = \frac{5x(2x^2)}{5x(3)} = \frac{2x^2}{3}$
$\frac{3y-9}{y-3} = \frac{3(y-3)}{(y-3)} = 3$
$\frac{5z^2+10z}{z+2} = \frac{5z(z+2)}{(z+2)} = 5z$

Explanation

This is like a mathematical magic trick where you make things disappear! By factoring, you reveal the secret identical parts of the top and bottom. Once you find a matching pair, you can cancel them out, leaving you with a much simpler, tidier expression. Presto, chango, simplified!

Property

Rewrite terms with negative exponents as fractions, such as $x^{-n} = \frac{1}{x^n}$. Once all terms have positive exponents and common denominators, combine them by adding or subtracting the numerators.

Examples

$\frac{4c}{d^2} - \frac{7c}{d^2} = \frac{4c-7c}{d^2} = -\frac{3c}{d^2}$
$xy^{-2} + \frac{5x}{y^2} = \frac{x}{y^2} + \frac{5x}{y^2} = \frac{x+5x}{y^2} = \frac{6x}{y^2}$

Explanation

Negative exponents are just shy and hiding in the wrong spot! A negative exponent in the numerator wants to be in the denominator. Move it to its happy place, and then you can combine terms that share the same denominator. It's all about getting everyone to the right party.