Pre-Algebra Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

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 $b = 5$ .

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

Step 3

Factor using the perfect square rule.

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Step 3.1

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

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

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 3.3

Rewrite the polynomial.

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

Step 3.4

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

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

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $x + 5$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x - 5$ .

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 4.3

Cancel the common factor of $x - 5$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

$1$