Simplify (x-2)/( square root of x^2-4)

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

Step 1

Simplify the denominator.

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

$\frac{x - 2}{\sqrt{x^{2} - 2^{2}}}$

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 $b = 2$ .

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

Step 2

Multiply $\frac{x - 2}{\sqrt{(x + 2) (x - 2)}}$ by $\frac{\sqrt{(x + 2) (x - 2)}}{\sqrt{(x + 2) (x - 2)}}$ .

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

Step 3

Combine and simplify the denominator.

Multiply $\frac{x - 2}{\sqrt{(x + 2) (x - 2)}}$ by $\frac{\sqrt{(x + 2) (x - 2)}}{\sqrt{(x + 2) (x - 2)}}$ .

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

Raise $\sqrt{(x + 2) (x - 2)}$ to the power of $1$ .

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

Raise $\sqrt{(x + 2) (x - 2)}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{(x + 2) (x - 2)}}^{2}$ as $(x + 2) (x - 2)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(x + 2) (x - 2)}$ as ${((x + 2) (x - 2))}^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 4

Cancel the common factor of $x - 2$ .

Cancel the common factor.

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

Rewrite the expression.

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