Simplify (4n)/( square root of 10n^5)

$\frac{4 n}{\sqrt{10 n^{5}}}$

Step 1

Simplify the denominator.

Rewrite $10 n^{5}$ as ${(n^{2})}^{2} \cdot (10 n)$ .

Factor out $n^{4}$ .

$\frac{4 n}{\sqrt{10 (n^{4} n)}}$

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

$\frac{4 n}{\sqrt{10 ({(n^{2})}^{2} n)}}$

Reorder $10$ and ${(n^{2})}^{2}$ .

$\frac{4 n}{\sqrt{{(n^{2})}^{2} \cdot 10 n}}$

Add parentheses.

$\frac{4 n}{\sqrt{{(n^{2})}^{2} \cdot (10 n)}}$

Pull terms out from under the radical.

$\frac{4 n}{n^{2} \sqrt{10 n}}$

Step 2

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

Factor $n$ out of $4 n$ .

$\frac{n \cdot 4}{n^{2} \sqrt{10 n}}$

Cancel the common factors.

Factor $n$ out of $n^{2} \sqrt{10 n}$ .

$\frac{n \cdot 4}{n (n \sqrt{10 n})}$

Cancel the common factor.

$\frac{n \cdot 4}{n (n \sqrt{10 n})}$

Rewrite the expression.

$\frac{4}{n \sqrt{10 n}}$

Step 3

Multiply $\frac{4}{n \sqrt{10 n}}$ by $\frac{\sqrt{10 n}}{\sqrt{10 n}}$ .

$\frac{4}{n \sqrt{10 n}} \cdot \frac{\sqrt{10 n}}{\sqrt{10 n}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{4}{n \sqrt{10 n}}$ by $\frac{\sqrt{10 n}}{\sqrt{10 n}}$ .

$\frac{4 \sqrt{10 n}}{n \sqrt{10 n} \sqrt{10 n}}$

Move $\sqrt{10 n}$ .

$\frac{4 \sqrt{10 n}}{n (\sqrt{10 n} \sqrt{10 n})}$

Raise $\sqrt{10 n}$ to the power of $1$ .

$\frac{4 \sqrt{10 n}}{n ({\sqrt{10 n}}^{1} \sqrt{10 n})}$

Raise $\sqrt{10 n}$ to the power of $1$ .

$\frac{4 \sqrt{10 n}}{n ({\sqrt{10 n}}^{1} {\sqrt{10 n}}^{1})}$

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

$\frac{4 \sqrt{10 n}}{n {\sqrt{10 n}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{4 \sqrt{10 n}}{n {\sqrt{10 n}}^{2}}$

Rewrite ${\sqrt{10 n}}^{2}$ as $10 n$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 n}$ as ${(10 n)}^{\frac{1}{2}}$ .

$\frac{4 \sqrt{10 n}}{n {({(10 n)}^{\frac{1}{2}})}^{2}}$

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

$\frac{4 \sqrt{10 n}}{n {(10 n)}^{\frac{1}{2} \cdot 2}}$

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

$\frac{4 \sqrt{10 n}}{n {(10 n)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 \sqrt{10 n}}{n {(10 n)}^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{4 \sqrt{10 n}}{n {(10 n)}^{1}}$

Simplify.

$\frac{4 \sqrt{10 n}}{n (10 n)}$

Step 5

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

Move $n$ .

$\frac{4 \sqrt{10 n}}{n \cdot n \cdot 10}$

Multiply $n$ by $n$ .

$\frac{4 \sqrt{10 n}}{n^{2} \cdot 10}$

Step 6

Cancel the common factor of $4$ and $10$ .

Factor $2$ out of $4 \sqrt{10 n}$ .

$\frac{2 (2 \sqrt{10 n})}{n^{2} \cdot 10}$

Cancel the common factors.

Factor $2$ out of $n^{2} \cdot 10$ .

$\frac{2 (2 \sqrt{10 n})}{2 (n^{2} \cdot 5)}$

Cancel the common factor.

$\frac{2 (2 \sqrt{10 n})}{2 (n^{2} \cdot 5)}$

Rewrite the expression.

$\frac{2 \sqrt{10 n}}{n^{2} \cdot 5}$

Step 7

Move $5$ to the left of $n^{2}$ .

$\frac{2 \sqrt{10 n}}{5 n^{2}}$

Simplify (4n)/( square root of 10n^5)

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