Simplify the Radical Expression ( fourth root of x)^-1

${(\sqrt[4]{x})}^{- 1}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{\sqrt[4]{x}}$

Multiply $\frac{1}{\sqrt[4]{x}}$ by $\frac{{\sqrt[4]{x}}^{3}}{{\sqrt[4]{x}}^{3}}$ .

$\frac{1}{\sqrt[4]{x}} \cdot \frac{{\sqrt[4]{x}}^{3}}{{\sqrt[4]{x}}^{3}}$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt[4]{x}}$ by $\frac{{\sqrt[4]{x}}^{3}}{{\sqrt[4]{x}}^{3}}$ .

$\frac{{\sqrt[4]{x}}^{3}}{\sqrt[4]{x} {\sqrt[4]{x}}^{3}}$

Raise $\sqrt[4]{x}$ to the power of $1$ .

$\frac{{\sqrt[4]{x}}^{3}}{{\sqrt[4]{x}}^{1} {\sqrt[4]{x}}^{3}}$

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

$\frac{{\sqrt[4]{x}}^{3}}{{\sqrt[4]{x}}^{1 + 3}}$

Add $1$ and $3$ .

$\frac{{\sqrt[4]{x}}^{3}}{{\sqrt[4]{x}}^{4}}$

Rewrite ${\sqrt[4]{x}}^{4}$ as $x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$\frac{{\sqrt[4]{x}}^{3}}{{(x^{\frac{1}{4}})}^{4}}$

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

$\frac{{\sqrt[4]{x}}^{3}}{x^{\frac{1}{4} \cdot 4}}$

Combine $\frac{1}{4}$ and $4$ .

$\frac{{\sqrt[4]{x}}^{3}}{x^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{{\sqrt[4]{x}}^{3}}{x^{\frac{4}{4}}}$

Rewrite the expression.

$\frac{{\sqrt[4]{x}}^{3}}{x^{1}}$

Simplify.

$\frac{{\sqrt[4]{x}}^{3}}{x}$

Rewrite ${\sqrt[4]{x}}^{3}$ as ${(x^{3})}^{\frac{1}{4}}$ .

$\frac{\sqrt[4]{x^{3}}}{x}$