Simplify (4n)/( square root of 18n^8)

$\frac{4 n}{\sqrt{18 n^{8}}}$

Simplify the denominator.

Rewrite $18 n^{8}$ as ${(3 n^{4})}^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$\frac{4 n}{\sqrt{9 (2) n^{8}}}$

Rewrite $9$ as $3^{2}$ .

$\frac{4 n}{\sqrt{3^{2} \cdot 2 n^{8}}}$

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

$\frac{4 n}{\sqrt{3^{2} \cdot 2 {(n^{4})}^{2}}}$

Move $2$ .

$\frac{4 n}{\sqrt{3^{2} {(n^{4})}^{2} \cdot 2}}$

Rewrite $3^{2} {(n^{4})}^{2}$ as ${(3 n^{4})}^{2}$ .

$\frac{4 n}{\sqrt{{(3 n^{4})}^{2} \cdot 2}}$

Pull terms out from under the radical.

$\frac{4 n}{3 n^{4} \sqrt{2}}$

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

Factor $n$ out of $4 n$ .

$\frac{n \cdot 4}{3 n^{4} \sqrt{2}}$

Cancel the common factors.

Factor $n$ out of $3 n^{4} \sqrt{2}$ .

$\frac{n \cdot 4}{n (3 n^{3} \sqrt{2})}$

Cancel the common factor.

$\frac{n \cdot 4}{n (3 n^{3} \sqrt{2})}$

Rewrite the expression.

$\frac{4}{3 n^{3} \sqrt{2}}$

Multiply $\frac{4}{3 n^{3} \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{4}{3 n^{3} \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Multiply $\frac{4}{3 n^{3} \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{4 \sqrt{2}}{3 n^{3} \sqrt{2} \sqrt{2}}$

Move $\sqrt{2}$ .

$\frac{4 \sqrt{2}}{3 n^{3} (\sqrt{2} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{4 \sqrt{2}}{3 n^{3} ({\sqrt{2}}^{1} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{4 \sqrt{2}}{3 n^{3} ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

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

$\frac{4 \sqrt{2}}{3 n^{3} {\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{4 \sqrt{2}}{3 n^{3} {\sqrt{2}}^{2}}$

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

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

$\frac{4 \sqrt{2}}{3 n^{3} {(2^{\frac{1}{2}})}^{2}}$

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

$\frac{4 \sqrt{2}}{3 n^{3} \cdot 2^{\frac{1}{2} \cdot 2}}$

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

$\frac{4 \sqrt{2}}{3 n^{3} \cdot 2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 \sqrt{2}}{3 n^{3} \cdot 2^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{4 \sqrt{2}}{3 n^{3} \cdot 2^{1}}$

Evaluate the exponent.

$\frac{4 \sqrt{2}}{3 n^{3} \cdot 2}$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4 \sqrt{2}$ .

$\frac{2 (2 \sqrt{2})}{3 n^{3} \cdot 2}$

Cancel the common factors.

Factor $2$ out of $3 n^{3} \cdot 2$ .

$\frac{2 (2 \sqrt{2})}{2 \cdot (3 n^{3})}$

Cancel the common factor.

$\frac{2 (2 \sqrt{2})}{2 \cdot (3 n^{3})}$

Rewrite the expression.

$\frac{2 \sqrt{2}}{3 n^{3}}$

Simplify (4n)/( square root of 18n^8)

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