1Cademy - Dividing $$\frac{3n^2}{n^2-4n} \div \frac{9n^2-45n}{n^2-7n+10}$$

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Rational Expression

Example

Dividing $\frac{3n^2}{n^2-4n} \div \frac{9n^2-45n}{n^2-7n+10}$

Divide $\frac{3n^2}{n^2-4n} \div \frac{9n^2-45n}{n^2-7n+10}$ .

Step 1 — Rewrite as multiplication by the reciprocal. Flip the second fraction and change division to multiplication:

$\frac{3n^2}{n^2-4n} \cdot \frac{n^2-7n+10}{9n^2-45n}$

Step 2 — Factor the numerators and denominators completely. Factor each polynomial: $3n^2 = 3 \cdot n \cdot n$ , $n^2 - 4n = n(n-4)$ , $n^2 - 7n + 10 = (n-5)(n-2)$ , and $9n^2 - 45n = 9n(n-5) = 3 \cdot 3 \cdot n(n-5)$ . The expression becomes:

$\frac{3 \cdot n \cdot n}{n(n-4)} \cdot \frac{(n-5)(n-2)}{3 \cdot 3 \cdot n(n-5)}$

Step 3 — Multiply the numerators and denominators:

$\frac{3 \cdot n \cdot n \cdot (n-5)(n-2)}{n(n-4) \cdot 3 \cdot 3 \cdot n(n-5)}$

Step 4 — Simplify by dividing out common factors. Cancel the common factors $3$ , $n$ , $n$ , and $(n-5)$ from both the numerator and denominator:

$\frac{n-2}{3(n-4)}$

This example demonstrates that after converting a division of rational expressions into a multiplication, the procedure is the same as multiplying rational expressions: factor completely, then cancel all shared factors between numerator and denominator.

Updated 2026-04-21

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OpenStax Elementary Algebra

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