Simplify ( square root of 8)/( square root of 3n^3)

$\frac{\sqrt{8}}{\sqrt{3 n^{3}}}$

Simplify the numerator.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical.

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

Simplify the denominator.

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

Factor out $n^{2}$ .

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

Reorder $3$ and $n^{2}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

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

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

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

Combine and simplify the denominator.

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

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

Move $\sqrt{3 n}$ .

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

Raise $\sqrt{3 n}$ to the power of $1$ .

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

Raise $\sqrt{3 n}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3 n}}^{2}$ as $3 n$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 n}$ as ${(3 n)}^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Multiply $n$ by $n$ by adding the exponents.

Move $n$ .

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

Multiply $n$ by $n$ .

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

$\frac{2 \sqrt{6 n}}{n^{2} \cdot 3}$

Move $3$ to the left of $n^{2}$ .

$\frac{2 \sqrt{6 n}}{3 n^{2}}$

Simplify ( square root of 8)/( square root of 3n^3)

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