Simplify ((4m^2-25n^2)/(m^3+8))÷((2m+5n)/(m^2-2m+4))

$\frac{4 m^{2} - 25 n^{2}}{m^{3} + 8} \div \frac{2 m + 5 n}{m^{2} - 2 m + 4}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{4 m^{2} - 25 n^{2}}{m^{3} + 8} \cdot \frac{m^{2} - 2 m + 4}{2 m + 5 n}$

Step 2

Simplify the numerator.

Rewrite $4 m^{2}$ as ${(2 m)}^{2}$ .

$\frac{{(2 m)}^{2} - 25 n^{2}}{m^{3} + 8} \cdot \frac{m^{2} - 2 m + 4}{2 m + 5 n}$

Rewrite $25 n^{2}$ as ${(5 n)}^{2}$ .

$\frac{{(2 m)}^{2} - {(5 n)}^{2}}{m^{3} + 8} \cdot \frac{m^{2} - 2 m + 4}{2 m + 5 n}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 m$ and $b = 5 n$ .

$\frac{(2 m + 5 n) (2 m - (5 n))}{m^{3} + 8} \cdot \frac{m^{2} - 2 m + 4}{2 m + 5 n}$

Multiply $5$ by $- 1$ .

$\frac{(2 m + 5 n) (2 m - 5 n)}{m^{3} + 8} \cdot \frac{m^{2} - 2 m + 4}{2 m + 5 n}$

Step 3

Simplify the denominator.

Rewrite $8$ as $2^{3}$ .

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

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = m$ and $b = 2$ .

$\frac{(2 m + 5 n) (2 m - 5 n)}{(m + 2) (m^{2} - m \cdot 2 + 2^{2})} \cdot \frac{m^{2} - 2 m + 4}{2 m + 5 n}$

Simplify.

Multiply $2$ by $- 1$ .

$\frac{(2 m + 5 n) (2 m - 5 n)}{(m + 2) (m^{2} - 2 m + 2^{2})} \cdot \frac{m^{2} - 2 m + 4}{2 m + 5 n}$

Raise $2$ to the power of $2$ .

$\frac{(2 m + 5 n) (2 m - 5 n)}{(m + 2) (m^{2} - 2 m + 4)} \cdot \frac{m^{2} - 2 m + 4}{2 m + 5 n}$

Step 4

Reduce the expression by cancelling the common factors.

Cancel the common factor of $2 m + 5 n$ .

Cancel the common factor.

$\frac{(2 m + 5 n) (2 m - 5 n)}{(m + 2) (m^{2} - 2 m + 4)} \cdot \frac{m^{2} - 2 m + 4}{2 m + 5 n}$

Rewrite the expression.

$\frac{2 m - 5 n}{(m + 2) (m^{2} - 2 m + 4)} (m^{2} - 2 m + 4)$

Cancel the common factor of $m^{2} - 2 m + 4$ .

Factor $m^{2} - 2 m + 4$ out of $(m + 2) (m^{2} - 2 m + 4)$ .

$\frac{2 m - 5 n}{(m^{2} - 2 m + 4) (m + 2)} (m^{2} - 2 m + 4)$

Cancel the common factor.

$\frac{2 m - 5 n}{(m^{2} - 2 m + 4) (m + 2)} (m^{2} - 2 m + 4)$

Rewrite the expression.

$\frac{2 m - 5 n}{m + 2}$

Simplify ((4m^2-25n^2)/(m^3+8))÷((2m+5n)/(m^2-2m+4))

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