Evaluate cube root of 5^4* fifth root of 5

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

Step 1

Raise $5$ to the power of $4$ .

$\sqrt[3]{625} \cdot \sqrt[5]{5}$

Step 2

Rewrite $625$ as $5^{3} \cdot 5$ .

Factor $125$ out of $625$ .

$\sqrt[3]{125 (5)} \cdot \sqrt[5]{5}$

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

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

Step 3

Pull terms out from under the radical.

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

Step 4

Multiply $5 \sqrt[3]{5} \sqrt[5]{5}$ .

Rewrite the expression using the least common index of $15$ .

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

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

Rewrite $5^{\frac{1}{5}}$ as $5^{\frac{3}{15}}$ .

$5 (5^{\frac{3}{15}} \sqrt[3]{5})$

Rewrite $5^{\frac{3}{15}}$ as $\sqrt[15]{5^{3}}$ .

$5 (\sqrt[15]{5^{3}} \sqrt[3]{5})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$5 (\sqrt[15]{5^{3}} \cdot 5^{\frac{1}{3}})$

Rewrite $5^{\frac{1}{3}}$ as $5^{\frac{5}{15}}$ .

$5 (\sqrt[15]{5^{3}} \cdot 5^{\frac{5}{15}})$

Rewrite $5^{\frac{5}{15}}$ as $\sqrt[15]{5^{5}}$ .

$5 (\sqrt[15]{5^{3}} \sqrt[15]{5^{5}})$

Combine using the product rule for radicals.

$5 \sqrt[15]{5^{3} \cdot 5^{5}}$

Multiply $5^{3}$ by $5^{5}$ by adding the exponents.

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

$5 \sqrt[15]{5^{3 + 5}}$

Add $3$ and $5$ .

$5 \sqrt[15]{5^{8}}$

Step 5

Raise $5$ to the power of $8$ .

$5 \sqrt[15]{390625}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$5 \sqrt[15]{390625}$

Decimal Form:

$11.79652267 \dots$