Find the Product square root of 13* fifth root of 13

$\sqrt{13} \cdot \sqrt[5]{13}$

Multiply $\sqrt{13} \sqrt[5]{13}$ .

Rewrite the expression using the least common index of $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$13^{\frac{1}{2}} \sqrt[5]{13}$

Rewrite $13^{\frac{1}{2}}$ as $13^{\frac{5}{10}}$ .

$13^{\frac{5}{10}} \sqrt[5]{13}$

Rewrite $13^{\frac{5}{10}}$ as $\sqrt[10]{13^{5}}$ .

$\sqrt[10]{13^{5}} \sqrt[5]{13}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{13}$ as $13^{\frac{1}{5}}$ .

$\sqrt[10]{13^{5}} \cdot 13^{\frac{1}{5}}$

Rewrite $13^{\frac{1}{5}}$ as $13^{\frac{2}{10}}$ .

$\sqrt[10]{13^{5}} \cdot 13^{\frac{2}{10}}$

Rewrite $13^{\frac{2}{10}}$ as $\sqrt[10]{13^{2}}$ .

$\sqrt[10]{13^{5}} \sqrt[10]{13^{2}}$

Combine using the product rule for radicals.

$\sqrt[10]{13^{5} \cdot 13^{2}}$

Multiply $13^{5}$ by $13^{2}$ by adding the exponents.

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

$\sqrt[10]{13^{5 + 2}}$

Add $5$ and $2$ .

$\sqrt[10]{13^{7}}$

Raise $13$ to the power of $7$ .

$\sqrt[10]{62748517}$

The result can be shown in multiple forms.

Exact Form:

$\sqrt[10]{62748517}$

Decimal Form:

$6.02227171 \dots$