Simplify seventh root of w square root of w

$\sqrt[7]{w} \sqrt{w}$

Step 1

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

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

$w^{\frac{1}{7}} \sqrt{w}$

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

$w^{\frac{2}{14}} \sqrt{w}$

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

$\sqrt[14]{w^{2}} \sqrt{w}$

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

$\sqrt[14]{w^{2}} w^{\frac{1}{2}}$

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

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

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

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

Step 2

Combine using the product rule for radicals.

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

Step 3

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

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

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

Add $2$ and $7$ .

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