Multiply Rational Expressions

Because rational expressions are just fractions with variables, the process of multiplying rational expressions is exactly the same as the process of multiplying fractions. Multiply the rational expressions just like you multiply fractions (numerator times numerator and denominator times denominator).

Just like with fractions, you can cross-cancel before you multiply.  So, if you have polynomials or polynomial expressions, factor them completely and cross-cancel any terms that you find in both a numerator and a denominator.  Remember you can cancel out terms attached with multiplication but if numbers or variable are attached by addition or subtraction they have to stay together.

Example

\require{cancel}\eqalign{ \dfrac{x^3+10x^2+25x}{x^2}\times\dfrac{x+1}{x^2-25}=&\\ \dfrac{\overset{x(x+5)(x+5)}{\bcancel{x^3+10x^2+25x}}}{x^2}\times\dfrac{x+1}{\underset{(x+5)(x-5)}{\bcancel{x-25}}}=& \quad&&\text{Factor completely}\\ \dfrac{\overset{\bcancel{x}\bcancel{(x+5)}(x+5)}{\bcancel{x^3+10x^2+25x}}}{\underset{\bcancel{x} \times x}{\bcancel{x^2}}}\times\dfrac{x+1}{\underset{\bcancel{(x+5)}(x-5)}{\bcancel{x^2-25}}}=& \quad&&\text{Cross cancel}\\ \dfrac{\overset{\bcancel{x}\bcancel{(x+5)}(x+5)}{\bcancel{x^3+10x^2+25x}}}{\underset{\bcancel{x}\times x}{\bcancel{x^2}}}\times\dfrac{x+1}{\underset{\bcancel{(x+5)}(x-5)}{\bcancel{x^2-25}}}=&\dfrac{(x+5)(x+1)}{x(x-5)}=\dfrac{x^2+6x+5}{x^2-5x}&&\text{Multiply numerator x numerator & denominator x denominator}}

Multiplying rational expressions is easy! Factor completely, cross cancel, and then multiply just like fractions.

• Multiply Rational Expressions

Simplify the rational expressions:

1. $\dfrac{x}{x^2}\times \dfrac{x}{x^2}$
2. $\dfrac{x}{3x}\times \dfrac{2}{x^2}$
3. $\dfrac{x+1}{x^2}\times \dfrac{3x}{x+1}$
4. $\dfrac{9x}{x^2+6x+9}\times \dfrac{x^2}{3x^2+12}$
5. $\dfrac{3x+3}{3x^3+6x}\times \dfrac{3x}{x+1}$
6. $\dfrac{x^2-x-12}{x^2+4x+4}\times \dfrac{x+2}{x-4}$
7. $\dfrac{x^3}{x^2+4x-45}\times \dfrac{x^2+9x}{x^2}$
8. $\dfrac{x^3-3x^2-40x}{6x}\times \dfrac{3x+12}{x^2-8x}$