Simplify fourth root of w sixth root of w^11

$\sqrt[4]{w} \sqrt[6]{w^{11}}$

Step 1

Factor out $w^{6}$ .

$\sqrt[4]{w} \sqrt[6]{w^{6} w^{5}}$

Step 2

Pull terms out from under the radical.

$\sqrt[4]{w} (w \sqrt[6]{w^{5}})$

Step 3

Multiply $\sqrt[4]{w} (w \sqrt[6]{w^{5}})$ .

Rewrite the expression using the least common index of $12$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{w}$ as $w^{\frac{1}{4}}$ .

$w (w^{\frac{1}{4}} \sqrt[6]{w^{5}})$

Rewrite $w^{\frac{1}{4}}$ as $w^{\frac{3}{12}}$ .

$w (w^{\frac{3}{12}} \sqrt[6]{w^{5}})$

Rewrite $w^{\frac{3}{12}}$ as $\sqrt[12]{w^{3}}$ .

$w (\sqrt[12]{w^{3}} \sqrt[6]{w^{5}})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{w^{5}}$ as $w^{\frac{5}{6}}$ .

$w (\sqrt[12]{w^{3}} w^{\frac{5}{6}})$

Rewrite $w^{\frac{5}{6}}$ as $w^{\frac{10}{12}}$ .

$w (\sqrt[12]{w^{3}} w^{\frac{10}{12}})$

Rewrite $w^{\frac{10}{12}}$ as $\sqrt[12]{w^{10}}$ .

$w (\sqrt[12]{w^{3}} \sqrt[12]{w^{10}})$

Combine using the product rule for radicals.

$w \sqrt[12]{w^{3} w^{10}}$

Multiply $w^{3}$ by $w^{10}$ by adding the exponents.

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

$w \sqrt[12]{w^{3 + 10}}$

Add $3$ and $10$ .

$w \sqrt[12]{w^{13}}$

Step 4

Factor out $w^{12}$ .

$w \sqrt[12]{w^{12} w}$

Step 5

Pull terms out from under the radical.

$w (w \sqrt[12]{w})$

Step 6

Multiply $w$ by $w$ by adding the exponents.

Move $w$ .

$w \cdot w \sqrt[12]{w}$

Multiply $w$ by $w$ .

$w^{2} \sqrt[12]{w}$