Find the Product square root of 11* ninth root of 11

$\sqrt{11} \cdot \sqrt[9]{11}$

Step 1

Multiply $\sqrt{11} \sqrt[9]{11}$ .

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$11^{\frac{1}{2}} \sqrt[9]{11}$

Rewrite $11^{\frac{1}{2}}$ as $11^{\frac{9}{18}}$ .

$11^{\frac{9}{18}} \sqrt[9]{11}$

Rewrite $11^{\frac{9}{18}}$ as $\sqrt[18]{11^{9}}$ .

$\sqrt[18]{11^{9}} \sqrt[9]{11}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{11}$ as $11^{\frac{1}{9}}$ .

$\sqrt[18]{11^{9}} \cdot 11^{\frac{1}{9}}$

Rewrite $11^{\frac{1}{9}}$ as $11^{\frac{2}{18}}$ .

$\sqrt[18]{11^{9}} \cdot 11^{\frac{2}{18}}$

Rewrite $11^{\frac{2}{18}}$ as $\sqrt[18]{11^{2}}$ .

$\sqrt[18]{11^{9}} \sqrt[18]{11^{2}}$

Combine using the product rule for radicals.

$\sqrt[18]{11^{9} \cdot 11^{2}}$

Multiply $11^{9}$ by $11^{2}$ by adding the exponents.

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

$\sqrt[18]{11^{9 + 2}}$

Add $9$ and $2$ .

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

Step 2

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

$\sqrt[18]{285311670611}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt[18]{285311670611}$

Decimal Form:

$4.32918994 \dots$