Simplify ( square root of x^2-1)/( square root of x-1)

$\frac{\sqrt{x^{2} - 1}}{\sqrt{x - 1}}$

Step 1

Simplify the numerator.

Rewrite $1$ as $1^{2}$ .

$\frac{\sqrt{x^{2} - 1^{2}}}{\sqrt{x - 1}}$

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 $b = 1$ .

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

Step 2

Combine $\sqrt{(x + 1) (x - 1)}$ and $\sqrt{x - 1}$ into a single radical.

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

Step 3

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{(x + 1) (x - 1)}{x - 1}$ by cancelling the common factors.

Cancel the common factor.

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

Rewrite the expression.

$\sqrt{\frac{x + 1}{1}}$

Divide $x + 1$ by $1$ .

$\sqrt{x + 1}$