Simplify ninth root of m square root of m

$\sqrt[9]{m} \sqrt{m}$

Step 1

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

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

$m^{\frac{1}{9}} \sqrt{m}$

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

$m^{\frac{2}{18}} \sqrt{m}$

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

$\sqrt[18]{m^{2}} \sqrt{m}$

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

$\sqrt[18]{m^{2}} m^{\frac{1}{2}}$

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

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

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

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

Step 2

Combine using the product rule for radicals.

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

Step 3

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

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

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

Add $2$ and $9$ .

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