Simplify fourth root of 5^3* square root of 5

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

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

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

Multiply $\sqrt[4]{125} \sqrt{5}$ .

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

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

$\sqrt[4]{125} \cdot 5^{\frac{1}{2}}$

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

$\sqrt[4]{125} \cdot 5^{\frac{2}{4}}$

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

$\sqrt[4]{125} \sqrt[4]{5^{2}}$

Combine using the product rule for radicals.

$\sqrt[4]{125 \cdot 5^{2}}$

Rewrite $125$ as $5^{3}$ .

$\sqrt[4]{5^{3} \cdot 5^{2}}$

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

$\sqrt[4]{5^{3 + 2}}$

Add $3$ and $2$ .

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

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

$\sqrt[4]{3125}$

Rewrite $3125$ as $5^{4} \cdot 5$ .

Factor $625$ out of $3125$ .

$\sqrt[4]{625 (5)}$

Rewrite $625$ as $5^{4}$ .

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

Pull terms out from under the radical.

$5 \sqrt[4]{5}$

The result can be shown in multiple forms.

Exact Form:

$5 \sqrt[4]{5}$

Decimal Form:

$7.47674390 \dots$