Simplify ( cube root of w* square root of w^5)/( square root of 25w^16)

$\frac{\sqrt[3]{w} \cdot \sqrt{w^{5}}}{\sqrt{25 w^{16}}}$

Step 1

Combine $\sqrt{w^{5}}$ and $\sqrt{25 w^{16}}$ into a single radical.

$\sqrt[3]{w} \cdot \sqrt{\frac{w^{5}}{25 w^{16}}}$

Step 2

Reduce the expression $\frac{w^{5}}{25 w^{16}}$ by cancelling the common factors.

Multiply by $1$ .

$\sqrt[3]{w} \cdot \sqrt{\frac{w^{5} \cdot 1}{25 w^{16}}}$

Factor $w^{5}$ out of $25 w^{16}$ .

$\sqrt[3]{w} \cdot \sqrt{\frac{w^{5} \cdot 1}{w^{5} (25 w^{11})}}$

Cancel the common factor.

$\sqrt[3]{w} \cdot \sqrt{\frac{w^{5} \cdot 1}{w^{5} (25 w^{11})}}$

Rewrite the expression.

$\sqrt[3]{w} \cdot \sqrt{\frac{1}{25 w^{11}}}$

Step 3

Rewrite $\frac{1}{25 w^{11}}$ as ${(\frac{1}{5 w^{5}})}^{2} \frac{1}{w}$ .

Factor the perfect power $1^{2}$ out of $1$ .

$\sqrt[3]{w} \cdot \sqrt{\frac{1^{2} \cdot 1}{25 w^{11}}}$

Factor the perfect power ${(5 w^{5})}^{2}$ out of $25 w^{11}$ .

$\sqrt[3]{w} \cdot \sqrt{\frac{1^{2} \cdot 1}{{(5 w^{5})}^{2} w}}$

Rearrange the fraction $\frac{1^{2} \cdot 1}{{(5 w^{5})}^{2} w}$ .

$\sqrt[3]{w} \cdot \sqrt{{(\frac{1}{5 w^{5}})}^{2} \frac{1}{w}}$

Step 4

Pull terms out from under the radical.

$\sqrt[3]{w} \cdot (\frac{1}{5 w^{5}} \sqrt{\frac{1}{w}})$

Step 5

Rewrite $\sqrt{\frac{1}{w}}$ as $\frac{\sqrt{1}}{\sqrt{w}}$ .

$\sqrt[3]{w} \cdot (\frac{1}{5 w^{5}} \cdot \frac{\sqrt{1}}{\sqrt{w}})$

Step 6

Combine.

$\sqrt[3]{w} \cdot \frac{1 \sqrt{1}}{5 w^{5} \sqrt{w}}$

Step 7

Multiply $\sqrt{1}$ by $1$ .

$\sqrt[3]{w} \cdot \frac{\sqrt{1}}{5 w^{5} \sqrt{w}}$

Step 8

Any root of $1$ is $1$ .

$\sqrt[3]{w} \cdot \frac{1}{5 w^{5} \sqrt{w}}$

Step 9

Multiply $\frac{1}{5 w^{5} \sqrt{w}}$ by $\frac{\sqrt{w}}{\sqrt{w}}$ .

$\sqrt[3]{w} \cdot (\frac{1}{5 w^{5} \sqrt{w}} \cdot \frac{\sqrt{w}}{\sqrt{w}})$

Step 10

Combine and simplify the denominator.

Multiply $\frac{1}{5 w^{5} \sqrt{w}}$ by $\frac{\sqrt{w}}{\sqrt{w}}$ .

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

Move $\sqrt{w}$ .

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

Raise $\sqrt{w}$ to the power of $1$ .

$\sqrt[3]{w} \cdot \frac{\sqrt{w}}{5 w^{5} ({\sqrt{w}}^{1} \sqrt{w})}$

Raise $\sqrt{w}$ to the power of $1$ .

$\sqrt[3]{w} \cdot \frac{\sqrt{w}}{5 w^{5} ({\sqrt{w}}^{1} {\sqrt{w}}^{1})}$

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

$\sqrt[3]{w} \cdot \frac{\sqrt{w}}{5 w^{5} {\sqrt{w}}^{1 + 1}}$

Add $1$ and $1$ .

$\sqrt[3]{w} \cdot \frac{\sqrt{w}}{5 w^{5} {\sqrt{w}}^{2}}$

Rewrite ${\sqrt{w}}^{2}$ as $w$ .

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

$\sqrt[3]{w} \cdot \frac{\sqrt{w}}{5 w^{5} {(w^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt[3]{w} \cdot \frac{\sqrt{w}}{5 w^{5} w^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\sqrt[3]{w} \cdot \frac{\sqrt{w}}{5 w^{5} w^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt[3]{w} \cdot \frac{\sqrt{w}}{5 w^{5} w^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt[3]{w} \cdot \frac{\sqrt{w}}{5 w^{5} w^{1}}$

Simplify.

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

Step 11

Simplify the denominator.

Raise $w$ to the power of $1$ .

$\sqrt[3]{w} \cdot \frac{\sqrt{w}}{5 (w^{1} w^{5})}$

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

$\sqrt[3]{w} \cdot \frac{\sqrt{w}}{5 w^{1 + 5}}$

Add $1$ and $5$ .

$\sqrt[3]{w} \cdot \frac{\sqrt{w}}{5 w^{6}}$

Step 12

Multiply $\sqrt[3]{w} \frac{\sqrt{w}}{5 w^{6}}$ .

Combine $\sqrt[3]{w}$ and $\frac{\sqrt{w}}{5 w^{6}}$ .

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

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 $w^{\frac{1}{3}}$ .

$\frac{w^{\frac{1}{3}} \sqrt{w}}{5 w^{6}}$

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

$\frac{w^{\frac{2}{6}} \sqrt{w}}{5 w^{6}}$

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

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

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

$\frac{\sqrt[6]{w^{2}} w^{\frac{1}{2}}}{5 w^{6}}$

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

$\frac{\sqrt[6]{w^{2}} w^{\frac{3}{6}}}{5 w^{6}}$

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

$\frac{\sqrt[6]{w^{2}} \sqrt[6]{w^{3}}}{5 w^{6}}$

Combine using the product rule for radicals.

$\frac{\sqrt[6]{w^{2} w^{3}}}{5 w^{6}}$

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

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

$\frac{\sqrt[6]{w^{2 + 3}}}{5 w^{6}}$

Add $2$ and $3$ .

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

Simplify ( cube root of w* square root of w^5)/( square root of 25w^16)

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