Evaluate square root of 5( fourth root of 5)

$\sqrt{5} (\sqrt[4]{5})$

Step 1

Multiply $\sqrt{5} (\sqrt[4]{5})$ .

Rewrite the expression using the least common index of $4$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$5^{\frac{1}{2}} \sqrt[4]{5}$

Rewrite $5^{\frac{1}{2}}$ as $5^{\frac{2}{4}}$ .

$5^{\frac{2}{4}} \sqrt[4]{5}$

Rewrite $5^{\frac{2}{4}}$ as $\sqrt[4]{5^{2}}$ .

$\sqrt[4]{5^{2}} \sqrt[4]{5}$

Combine using the product rule for radicals.

$\sqrt[4]{5^{2} \cdot 5}$

Multiply $5^{2}$ by $5$ by adding the exponents.

Multiply $5^{2}$ by $5$ .

Raise $5$ to the power of $1$ .

$\sqrt[4]{5^{2} \cdot 5^{1}}$

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

$\sqrt[4]{5^{2 + 1}}$

Add $2$ and $1$ .

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

Step 2

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

$\sqrt[4]{125}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt[4]{125}$

Decimal Form:

$3.34370152 \dots$