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

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

Step 1

Simplify the denominator.

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

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

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 $b = 1$ .

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{(x + 1) (x - 1)}}^{2}$ as $(x + 1) (x - 1)$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 4

Cancel the common factor of $x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

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