Simplify ((x^2-25)/(x^2-10x+25))/((x+5)/(x-5))

$\frac{\frac{x^{2} - 25}{x^{2} - 10 x + 25}}{\frac{x + 5}{x - 5}}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} - 25}{x^{2} - 10 x + 25} \cdot \frac{x - 5}{x + 5}$

Simplify the numerator.

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Rewrite $25$ as $5^{2}$ .

$\frac{x^{2} - 5^{2}}{x^{2} - 10 x + 25} \cdot \frac{x - 5}{x + 5}$

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

$\frac{(x + 5) (x - 5)}{x^{2} - 10 x + 25} \cdot \frac{x - 5}{x + 5}$

Factor using the perfect square rule.

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Rewrite $25$ as $5^{2}$ .

$\frac{(x + 5) (x - 5)}{x^{2} - 10 x + 5^{2}} \cdot \frac{x - 5}{x + 5}$

Check the middle term by multiplying $2 a b$ and compare this result with the middle term in the original expression.

$2 a b = 2 \cdot x \cdot - 5$

Simplify.

$2 a b = - 10 x$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = - 5$ .

$\frac{(x + 5) (x - 5)}{{(x - 5)}^{2}} \cdot \frac{x - 5}{x + 5}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor of $x + 5$ .

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Cancel the common factor.

$\frac{(x + 5) (x - 5)}{{(x - 5)}^{2}} \cdot \frac{x - 5}{x + 5}$

Rewrite the expression.

$\frac{x - 5}{{(x - 5)}^{2}} (x - 5)$

Cancel the common factor of $x - 5$ .

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Factor $x - 5$ out of ${(x - 5)}^{2}$ .

$\frac{x - 5}{(x - 5) (x - 5)} (x - 5)$

Cancel the common factor.

$\frac{x - 5}{(x - 5) (x - 5)} (x - 5)$

Rewrite the expression.

$\frac{x - 5}{x - 5}$

Cancel the common factor of $x - 5$ .

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Cancel the common factor.

$\frac{x - 5}{x - 5}$

Divide $1$ by $1$ .

$1$

Simplify ((x^2-25)/(x^2-10x+25))/((x+5)/(x-5))

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