Simplify ( square root of m)/( square root of 9n^3)

$\frac{\sqrt{m}}{\sqrt{9 n^{3}}}$

Simplify the denominator.

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Rewrite $9 n^{3}$ as ${(3 n)}^{2} n$ .

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

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

Factor out $n^{2}$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Combine and simplify the denominator.

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Multiply $\frac{\sqrt{m}}{3 n \sqrt{n}}$ and $\frac{\sqrt{n}}{\sqrt{n}}$ .

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

Move $\sqrt{n}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $1$ by $1$ .

$\frac{\sqrt{m} \sqrt{n}}{3 n \cdot n^{1}}$

Simplify.

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

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

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Move $n$ .

$\frac{\sqrt{m} \sqrt{n}}{3 (n \cdot n)}$

Multiply $n$ by $n$ .

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

Combine using the product rule for radicals.

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

Simplify ( square root of m)/( square root of 9n^3)