Simplify square root of 13( sixth root of 13)

$\sqrt{13} (\sqrt[6]{13})$

Step 1

Multiply $\sqrt{13} (\sqrt[6]{13})$ .

Rewrite the expression using the least common index of $6$ .

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

$13^{\frac{1}{2}} \sqrt[6]{13}$

Rewrite $13^{\frac{1}{2}}$ as $13^{\frac{3}{6}}$ .

$13^{\frac{3}{6}} \sqrt[6]{13}$

Rewrite $13^{\frac{3}{6}}$ as $\sqrt[6]{13^{3}}$ .

$\sqrt[6]{13^{3}} \sqrt[6]{13}$

Combine using the product rule for radicals.

$\sqrt[6]{13^{3} \cdot 13}$

Multiply $13^{3}$ by $13$ by adding the exponents.

Multiply $13^{3}$ by $13$ .

Raise $13$ to the power of $1$ .

$\sqrt[6]{13^{3} \cdot 13^{1}}$

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

$\sqrt[6]{13^{3 + 1}}$

Add $3$ and $1$ .

$\sqrt[6]{13^{4}}$

Step 2

Raise $13$ to the power of $4$ .

$\sqrt[6]{28561}$

Step 3

Rewrite $28561$ as $169^{2}$ .

$\sqrt[6]{169^{2}}$

Step 4

Rewrite $\sqrt[6]{169^{2}}$ as $\sqrt[3]{\sqrt{169^{2}}}$ .

$\sqrt[3]{\sqrt{169^{2}}}$

Step 5

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt[3]{169}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt[3]{169}$

Decimal Form:

$5.52877481 \dots$