1Cademy - Simplifying $$\frac{\frac{4}{y-3}}{\frac{8}{y^2-9}}$$

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Procedure to Simplify a Complex Rational Expression by Writing it as Division

Example

Simplifying $\frac{\frac{4}{y-3}}{\frac{8}{y^2-9}}$

Simplify the complex rational expression:

$\frac{\frac{4}{y-3}}{\frac{8}{y^2-9}}$

Step 1 — Rewrite the complex fraction as division. The main fraction bar represents division, so rewrite the expression as the numerator fraction divided by the denominator fraction:

$\frac{4}{y-3} \div \frac{8}{y^2-9}$

Step 2 — Multiply by the reciprocal of the second fraction:

$\frac{4}{y-3} \cdot \frac{y^2-9}{8}$

Step 3 — Multiply the numerators and denominators:

$\frac{4(y^2-9)}{8(y-3)}$

Step 4 — Factor to reveal common factors. Recognize $y^2 - 9$ as a difference of squares: $y^2 - 9 = (y-3)(y+3)$ . Factor $8$ as $4 \cdot 2$ :

$\frac{4(y-3)(y+3)}{4 \cdot 2(y-3)}$

Step 5 — Cancel the common factors $4$ and $(y-3)$ :

$\frac{y+3}{2}$

Although the simplified expression $\frac{y+3}{2}$ has only a constant in the denominator, the original complex rational expression contained the denominators $y - 3$ and $y^2 - 9$ . Setting each equal to zero shows that $y = 3$ and $y = -3$ would make the original expression undefined, so both values must be excluded even though they do not appear in the simplified form.

Updated 2026-04-21

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OpenStax Elementary Algebra

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