Simplify ((x^2-4)/x)/((x+2)/(x-2))

$\frac{\frac{x^{2} - 4}{x}}{\frac{x + 2}{x - 2}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} - 4}{x} \cdot \frac{x - 2}{x + 2}$

Step 2

Simplify the numerator.

Rewrite $4$ as $2^{2}$ .

$\frac{x^{2} - 2^{2}}{x} \cdot \frac{x - 2}{x + 2}$

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 $b = 2$ .

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

Step 3

Simplify terms.

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{x - 2}{x}$ by $x - 2$ .

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

Step 4

Simplify the numerator.

Raise $x - 2$ to the power of $1$ .

$\frac{{(x - 2)}^{1} (x - 2)}{x}$

Raise $x - 2$ to the power of $1$ .

$\frac{{(x - 2)}^{1} {(x - 2)}^{1}}{x}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(x - 2)}^{1 + 1}}{x}$

Add $1$ and $1$ .

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