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

$\sqrt{13} \cdot \sqrt[7]{13}$

Multiply $\sqrt{13} \sqrt[7]{13}$ .

Rewrite the expression using the least common index of $14$ .

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

$13^{\frac{1}{2}} \sqrt[7]{13}$

Rewrite $13^{\frac{1}{2}}$ as $13^{\frac{7}{14}}$ .

$13^{\frac{7}{14}} \sqrt[7]{13}$

Rewrite $13^{\frac{7}{14}}$ as $\sqrt[14]{13^{7}}$ .

$\sqrt[14]{13^{7}} \sqrt[7]{13}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{13}$ as $13^{\frac{1}{7}}$ .

$\sqrt[14]{13^{7}} \cdot 13^{\frac{1}{7}}$

Rewrite $13^{\frac{1}{7}}$ as $13^{\frac{2}{14}}$ .

$\sqrt[14]{13^{7}} \cdot 13^{\frac{2}{14}}$

Rewrite $13^{\frac{2}{14}}$ as $\sqrt[14]{13^{2}}$ .

$\sqrt[14]{13^{7}} \sqrt[14]{13^{2}}$

Combine using the product rule for radicals.

$\sqrt[14]{13^{7} \cdot 13^{2}}$

Multiply $13^{7}$ by $13^{2}$ by adding the exponents.

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

$\sqrt[14]{13^{7 + 2}}$

Add $7$ and $2$ .

$\sqrt[14]{13^{9}}$

Raise $13$ to the power of $9$ .

$\sqrt[14]{10604499373}$

The result can be shown in multiple forms.

Exact Form:

$\sqrt[14]{10604499373}$

Decimal Form:

$5.20123457 \dots$