Simplify (5w^2)/( square root of 30w^9)

$\frac{5 w^{2}}{\sqrt{30 w^{9}}}$

Step 1

Simplify the denominator.

Rewrite $30 w^{9}$ as ${({(w^{2})}^{2})}^{2} \cdot (30 w)$ .

Factor out $w^{8}$ .

$\frac{5 w^{2}}{\sqrt{30 (w^{8} w)}}$

Rewrite $w^{8}$ as ${(w^{4})}^{2}$ .

$\frac{5 w^{2}}{\sqrt{30 ({(w^{4})}^{2} w)}}$

Reorder $30$ and ${(w^{4})}^{2}$ .

$\frac{5 w^{2}}{\sqrt{{(w^{4})}^{2} \cdot 30 w}}$

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

$\frac{5 w^{2}}{\sqrt{{({(w^{2})}^{2})}^{2} \cdot 30 w}}$

Add parentheses.

$\frac{5 w^{2}}{\sqrt{{({(w^{2})}^{2})}^{2} \cdot (30 w)}}$

Pull terms out from under the radical.

$\frac{5 w^{2}}{{(w^{2})}^{2} \sqrt{30 w}}$

Multiply the exponents in ${(w^{2})}^{2}$ .

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

$\frac{5 w^{2}}{w^{2 \cdot 2} \sqrt{30 w}}$

Multiply $2$ by $2$ .

$\frac{5 w^{2}}{w^{4} \sqrt{30 w}}$

Step 2

Cancel the common factor of $w^{2}$ and $w^{4}$ .

Factor $w^{2}$ out of $5 w^{2}$ .

$\frac{w^{2} \cdot 5}{w^{4} \sqrt{30 w}}$

Cancel the common factors.

Factor $w^{2}$ out of $w^{4} \sqrt{30 w}$ .

$\frac{w^{2} \cdot 5}{w^{2} (w^{2} \sqrt{30 w})}$

Cancel the common factor.

$\frac{w^{2} \cdot 5}{w^{2} (w^{2} \sqrt{30 w})}$

Rewrite the expression.

$\frac{5}{w^{2} \sqrt{30 w}}$

Step 3

Multiply $\frac{5}{w^{2} \sqrt{30 w}}$ by $\frac{\sqrt{30 w}}{\sqrt{30 w}}$ .

$\frac{5}{w^{2} \sqrt{30 w}} \cdot \frac{\sqrt{30 w}}{\sqrt{30 w}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{5}{w^{2} \sqrt{30 w}}$ by $\frac{\sqrt{30 w}}{\sqrt{30 w}}$ .

$\frac{5 \sqrt{30 w}}{w^{2} \sqrt{30 w} \sqrt{30 w}}$

Move $\sqrt{30 w}$ .

$\frac{5 \sqrt{30 w}}{w^{2} (\sqrt{30 w} \sqrt{30 w})}$

Raise $\sqrt{30 w}$ to the power of $1$ .

$\frac{5 \sqrt{30 w}}{w^{2} ({\sqrt{30 w}}^{1} \sqrt{30 w})}$

Raise $\sqrt{30 w}$ to the power of $1$ .

$\frac{5 \sqrt{30 w}}{w^{2} ({\sqrt{30 w}}^{1} {\sqrt{30 w}}^{1})}$

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

$\frac{5 \sqrt{30 w}}{w^{2} {\sqrt{30 w}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{5 \sqrt{30 w}}{w^{2} {\sqrt{30 w}}^{2}}$

Rewrite ${\sqrt{30 w}}^{2}$ as $30 w$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{30 w}$ as ${(30 w)}^{\frac{1}{2}}$ .

$\frac{5 \sqrt{30 w}}{w^{2} {({(30 w)}^{\frac{1}{2}})}^{2}}$

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

$\frac{5 \sqrt{30 w}}{w^{2} {(30 w)}^{\frac{1}{2} \cdot 2}}$

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

$\frac{5 \sqrt{30 w}}{w^{2} {(30 w)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{5 \sqrt{30 w}}{w^{2} {(30 w)}^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{5 \sqrt{30 w}}{w^{2} {(30 w)}^{1}}$

Simplify.

$\frac{5 \sqrt{30 w}}{w^{2} (30 w)}$

Step 5

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

Move $w$ .

$\frac{5 \sqrt{30 w}}{w \cdot w^{2} \cdot 30}$

Multiply $w$ by $w^{2}$ .

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

$\frac{5 \sqrt{30 w}}{w^{1} w^{2} \cdot 30}$

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

$\frac{5 \sqrt{30 w}}{w^{1 + 2} \cdot 30}$

Add $1$ and $2$ .

$\frac{5 \sqrt{30 w}}{w^{3} \cdot 30}$

Step 6

Cancel the common factor of $5$ and $30$ .

Factor $5$ out of $5 \sqrt{30 w}$ .

$\frac{5 (\sqrt{30 w})}{w^{3} \cdot 30}$

Cancel the common factors.

Factor $5$ out of $w^{3} \cdot 30$ .

$\frac{5 (\sqrt{30 w})}{5 (w^{3} \cdot 6)}$

Cancel the common factor.

$\frac{5 \sqrt{30 w}}{5 (w^{3} \cdot 6)}$

Rewrite the expression.

$\frac{\sqrt{30 w}}{w^{3} \cdot 6}$

Step 7

Move $6$ to the left of $w^{3}$ .

$\frac{\sqrt{30 w}}{6 w^{3}}$

Simplify (5w^2)/( square root of 30w^9)

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