Multiplying Rational Equations

Just like with numerical ratios, products of rational expressions are found by multiplying the numerators together and the denominators together seperately. When multiplying rational expressions, divide out common factors to simplify the product.

Example 1:

\frac{8 x^{3} y}{4 x y^{2}} \times \frac{7 x^{3} y^{2}}{2 x y^{3}}

\frac{8 x x x y 7 x x x y y}{4 x y y 2 x y y y}

Multiply straight across and expand the factors of the numerator and denominator.

\frac{8 x x x y 7 x x x y y}{4 x y y 2 x y y y}

Simplify by eliminating factors that are in both the numerator and denominator.

\frac{7 x^{4}}{y^{2}}

Rewrite the simplified form.

Example 2:

\frac{3 x^{2} - 9 x}{x^{2} - 4 x - 5} \times \frac{x^{2} - 4 x - 5}{6 x}

\frac{3 x (x - 3) \times (x - 5) (x + 1)}{(x - 5) (x + 1) \times 2 • 3 x}

Multiply and factor the numerator and denominator.

\frac{3 x (x - 3) \times (x - 5) (x + 1)}{(x - 5) (x + 1) \times 2 • 3 x}

Eliminate factors that are in both the numerator and denominator.

\frac{x - 3}{2}

Rewrite.

Example 3:

\frac{4 x + 20}{2 x + 2} \times \frac{x^{2} + x}{4 x^{2} + 20 x}

\frac{4 (x + 5) \times x (x + 1)}{2 (x + 1) \times 4 x (x + 5)}

Multiply and factor the numerator and denominator.

\frac{4 (x + 5) \times x (x + 1)}{2 (x + 1) \times 4 x (x + 5)}

Eliminate factors that are in both the numerator and denominator.

\frac{1}{2}

Rewrite.

Note: When dividing rational expressions, simply use the reciprocal of the rational divisor and the operation becomes multiplication.

Example 4:

\frac{5 x^{2} y}{3 x y^{3}} \div \frac{35 x^{2} y}{42 x y^{2}}

\frac{5 x^{2} y}{3 x y^{3}} \times \frac{42 x y^{2}}{35 x^{2} y}

Rewrite as a multiplication expression by using the reciprocal of the divisor.

\frac{5 x x y}{3 x y y y} \frac{2 • 3 • 7 x y y}{5 • 7 x x y}

Expand the factors of the numerator and denominator.

\frac{5 x x y}{3 x y y y} \frac{2 • 3 • 7 x y y}{5 • 7 x x y}

Eliminate factors that are in both the numerator and denominator.

\frac{2}{y}

Rewrite.

Example 5:

\frac{x^{2} + 12 x + 32}{4 x + 28} \div \frac{x^{2} + 4 x}{x^{2} - 49}

\frac{x^{2} + 12 x + 32}{4 x + 28} \times \frac{x^{2} - 49}{x^{2} + 4 x}

Rewrite as a mutiplication expression.

\frac{(x + 4) (x + 8)}{4 (x + 7)} \times \frac{(x - 7) (x + 7)}{x (x + 4)}

Factor the numerator and denominator.

\frac{(x + 4) (x + 8)}{4 (x + 7)} \frac{(x - 7) (x + 7)}{x (x + 4)}

Eliminate factors that are in both the numerator and denominator.

\frac{(x + 8) (x + 7)}{4 x}

Rewrite.

Example 6:

\frac{x^{2} - 3 x - 10}{x^{2} + 4 x + 3} \div (x^{2} + x - 2)

\frac{x^{2} - x - 6}{x^{2} - 2 x - 3} \times \frac{1}{x^{2} + x - 2}

Rewrite as a multiplication expression.

\frac{(x - 3) (x + 2)}{(x + 1) (x - 3)} \times \frac{1}{(x + 2) (x - 1)}

Factor the numerator and denominator.

\frac{(x - 3) (x + 2)}{(x + 1) (x - 3)} \frac{1}{(x + 2) (x - 1)}

Eliminate factors that are in both the numerator and denominator.

\frac{1}{(x + 1) (x - 1)}

Rewrite.

Now that you can multiply and divide rational expressions you are one step closer to solving rational equations.

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