Simplify square root of 1/(x-1)

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

Rewrite $\sqrt{\frac{1}{x - 1}}$ as $\frac{\sqrt{1}}{\sqrt{x - 1}}$ .

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

Any root of $1$ is $1$ .

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

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

$\frac{1}{\sqrt{x - 1}} \cdot \frac{\sqrt{x - 1}}{\sqrt{x - 1}}$

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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