Simplify cube root of w* square root of w^5

$\sqrt[3]{w} \cdot \sqrt{w^{5}}$

Step 1

Rewrite $w^{5}$ as ${(w^{2})}^{2} w$ .

Factor out $w^{4}$ .

$\sqrt[3]{w} \cdot \sqrt{w^{4} w}$

Rewrite $w^{4}$ as ${(w^{2})}^{2}$ .

$\sqrt[3]{w} \cdot \sqrt{{(w^{2})}^{2} w}$

Step 2

Pull terms out from under the radical.

$\sqrt[3]{w} \cdot (w^{2} \sqrt{w})$

Step 3

Multiply $\sqrt[3]{w} (w^{2} \sqrt{w})$ .

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{w}$ as $w^{\frac{1}{3}}$ .

$w^{2} (w^{\frac{1}{3}} \sqrt{w})$

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

$w^{2} (w^{\frac{2}{6}} \sqrt{w})$

Rewrite $w^{\frac{2}{6}}$ as $\sqrt[6]{w^{2}}$ .

$w^{2} (\sqrt[6]{w^{2}} \sqrt{w})$

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

$w^{2} (\sqrt[6]{w^{2}} w^{\frac{1}{2}})$

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

$w^{2} (\sqrt[6]{w^{2}} w^{\frac{3}{6}})$

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

$w^{2} (\sqrt[6]{w^{2}} \sqrt[6]{w^{3}})$

Combine using the product rule for radicals.

$w^{2} \sqrt[6]{w^{2} w^{3}}$

Multiply $w^{2}$ by $w^{3}$ by adding the exponents.

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

$w^{2} \sqrt[6]{w^{2 + 3}}$

Add $2$ and $3$ .

$w^{2} \sqrt[6]{w^{5}}$