To perform multiple operations on rational expressions, such as $$\frac{3x-6}{4x-4} \cdot \frac{x^2+2x-3}{x^2-3x-10} \div \frac{2x+12}{8x+16}$$, first rewrite the division as multiplication by the reciprocal: $$\frac{3x-6}{4x-4} \cdot \frac{x^2+2x-3}{x^2-3x-10} \cdot \frac{8x+16}{2x+12}$$. Next, factor all numerators and denominators to obtain $$\frac{3(x-2)}{4(x-1)} \cdot \frac{(x+3)(x-1)}{(x-5)(x+2)} \cdot \frac{8(x+2)}{2(x+6)}$$. Then, multiply the fractions together, bringing constants to the front to help identify numerical factors: $$\frac{3 \cdot 8(x-2)(x+3)(x-1)(x+2)}{4 \cdot 2(x-1)(x-5)(x+2)(x+6)}$$. Finally, simplify by dividing out the common factors of $$8$$, $$(x-1)$$, and $$(x+2)$$ from both the numerator and denominator, resulting in the simplified expression $$\frac{3(x-2)(x+3)}{(x-5)(x+6)}$$.

Example of Multiplying and Dividing Multiple Rational Expressions

When an expression contains more than two rational expressions connected by multiplication and division, the standard order of operations and procedures still apply. The first step is to rewrite any division operation as multiplication by the reciprocal of the rational expression immediately following the division sign. After converting all operations to multiplication, factor all numerators and denominators completely, and then multiply the expressions together before simplifying the final result.

Claude

To multiply rational expressions, use the following steps:
1. Factor each numerator and denominator completely.
2. Multiply the numerators and denominators.
3. Simplify the result by dividing out any common factors.

Procedure for Multiplying Rational Expressions

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OpenStax Intermediate Algebra

An example of multiplying rational expressions is simplifying $$\frac{2x}{x^2 - 7x + 12} \cdot \frac{x^2 - 9}{6x^2}$$.

Step 1. Factor each numerator and denominator completely: $$\frac{2x}{(x - 3)(x - 4)} \cdot \frac{(x - 3)(x + 3)}{6x^2}$$.
Step 2. Multiply the numerators and denominators: $$\frac{2x(x - 3)(x + 3)}{6x^2(x - 3)(x - 4)}$$.
Step 3. Simplify by dividing out the common factors $$(x - 3)$$ and $$2x$$. The simplified result is $$\frac{x + 3}{3x(x - 4)}$$.

Example of Multiplying Rational Expressions

To multiply rational expressions involving trinomials, such as $$\frac{3a^2-8a-3}{a^2-25} \cdot \frac{a^2+10a+25}{3a^2-14a-5}$$, first factor all numerators and denominators completely to obtain $$\frac{(3a+1)(a-3)(a+5)(a+5)}{(a-5)(a+5)(3a+1)(a-5)}$$. Then, simplify the expression by dividing out the common factors $$(3a+1)$$ and $$(a+5)$$, which leaves $$\frac{(a-3)(a+5)}{(a-5)(a-5)}$$. Finally, rewrite any repeated factors using an exponent, resulting in the simplified product $$\frac{(a-3)(a+5)}{(a-5)^2}$$.

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