Simplify x/( square root of x-x)

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

Simplify the denominator.

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

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

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} - x$ .

Multiply by $1$ .

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

Factor $x^{\frac{1}{2}}$ out of $- x$ .

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

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} \cdot 1 + x^{\frac{1}{2}} (- x^{\frac{1}{2}})$ .

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

Move $x^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

Multiply $x$ by $x^{- \frac{1}{2}}$ .

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

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Subtract $1$ from $2$ .

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