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

$\sqrt{13} \cdot \sqrt[9]{13}$

Multiply $\sqrt{13} \sqrt[9]{13}$ .

Rewrite the expression using the least common index of $18$ .

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

$13^{\frac{1}{2}} \sqrt[9]{13}$

Rewrite $13^{\frac{1}{2}}$ as $13^{\frac{9}{18}}$ .

$13^{\frac{9}{18}} \sqrt[9]{13}$

Rewrite $13^{\frac{9}{18}}$ as $\sqrt[18]{13^{9}}$ .

$\sqrt[18]{13^{9}} \sqrt[9]{13}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{13}$ as $13^{\frac{1}{9}}$ .

$\sqrt[18]{13^{9}} \cdot 13^{\frac{1}{9}}$

Rewrite $13^{\frac{1}{9}}$ as $13^{\frac{2}{18}}$ .

$\sqrt[18]{13^{9}} \cdot 13^{\frac{2}{18}}$

Rewrite $13^{\frac{2}{18}}$ as $\sqrt[18]{13^{2}}$ .

$\sqrt[18]{13^{9}} \sqrt[18]{13^{2}}$

Combine using the product rule for radicals.

$\sqrt[18]{13^{9} \cdot 13^{2}}$

Multiply $13^{9}$ by $13^{2}$ by adding the exponents.

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

$\sqrt[18]{13^{9 + 2}}$

Add $9$ and $2$ .

$\sqrt[18]{13^{11}}$

Raise $13$ to the power of $11$ .

$\sqrt[18]{1792160394037}$

The result can be shown in multiple forms.

Exact Form:

$\sqrt[18]{1792160394037}$

Decimal Form:

$4.79449819 \dots$