Simplify the Radical Expression 1/( fourth root of 5)

$\frac{1}{\sqrt[4]{5}}$

Step 1

Multiply $\frac{1}{\sqrt[4]{5}}$ by $\frac{{\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{3}}$ .

$\frac{1}{\sqrt[4]{5}} \cdot \frac{{\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{3}}$

Step 2

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt[4]{5}}$ by $\frac{{\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{3}}$ .

$\frac{{\sqrt[4]{5}}^{3}}{\sqrt[4]{5} {\sqrt[4]{5}}^{3}}$

Raise $\sqrt[4]{5}$ to the power of $1$ .

$\frac{{\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{1} {\sqrt[4]{5}}^{3}}$

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

$\frac{{\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{1 + 3}}$

Add $1$ and $3$ .

$\frac{{\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{4}}$

Rewrite ${\sqrt[4]{5}}^{4}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{5}$ as $5^{\frac{1}{4}}$ .

$\frac{{\sqrt[4]{5}}^{3}}{{(5^{\frac{1}{4}})}^{4}}$

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

$\frac{{\sqrt[4]{5}}^{3}}{5^{\frac{1}{4} \cdot 4}}$

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

$\frac{{\sqrt[4]{5}}^{3}}{5^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{{\sqrt[4]{5}}^{3}}{5^{\frac{4}{4}}}$

Rewrite the expression.

$\frac{{\sqrt[4]{5}}^{3}}{5^{1}}$

Evaluate the exponent.

$\frac{{\sqrt[4]{5}}^{3}}{5}$

Step 3

Simplify the numerator.

Rewrite ${\sqrt[4]{5}}^{3}$ as ${(5^{3})}^{\frac{1}{4}}$ .

$\frac{\sqrt[4]{5^{3}}}{5}$

Raise $5$ to the power of $3$ .

$\frac{\sqrt[4]{125}}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[4]{125}}{5}$

Decimal Form:

$0.66874030 \dots$