Simplify (2 square root of 3n)/(2 square root of 5n^3)

$\frac{2 \sqrt{3 n}}{2 \sqrt{5 n^{3}}}$

Step 1

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \sqrt{3 n}}{2 \sqrt{5 n^{3}}}$

Rewrite the expression.

$\frac{\sqrt{3 n}}{\sqrt{5 n^{3}}}$

Step 2

Combine $\sqrt{3 n}$ and $\sqrt{5 n^{3}}$ into a single radical.

$\sqrt{\frac{3 n}{5 n^{3}}}$

Step 3

Reduce the expression $\frac{3 n}{5 n^{3}}$ by cancelling the common factors.

Factor $n$ out of $3 n$ .

$\sqrt{\frac{n \cdot 3}{5 n^{3}}}$

Factor $n$ out of $5 n^{3}$ .

$\sqrt{\frac{n \cdot 3}{n (5 n^{2})}}$

Cancel the common factor.

$\sqrt{\frac{n \cdot 3}{n (5 n^{2})}}$

Rewrite the expression.

$\sqrt{\frac{3}{5 n^{2}}}$

Step 4

Rewrite $\frac{3}{5 n^{2}}$ as ${(\frac{1}{n})}^{2} \frac{3}{5}$ .

Factor the perfect power $1^{2}$ out of $3$ .

$\sqrt{\frac{1^{2} \cdot 3}{5 n^{2}}}$

Factor the perfect power $n^{2}$ out of $5 n^{2}$ .

$\sqrt{\frac{1^{2} \cdot 3}{n^{2} \cdot 5}}$

Rearrange the fraction $\frac{1^{2} \cdot 3}{n^{2} \cdot 5}$ .

$\sqrt{{(\frac{1}{n})}^{2} \frac{3}{5}}$

Step 5

Pull terms out from under the radical.

$\frac{1}{n} \sqrt{\frac{3}{5}}$

Step 6

Rewrite $\sqrt{\frac{3}{5}}$ as $\frac{\sqrt{3}}{\sqrt{5}}$ .

$\frac{1}{n} \cdot \frac{\sqrt{3}}{\sqrt{5}}$

Step 7

Multiply $\frac{\sqrt{3}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{1}{n} (\frac{\sqrt{3}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 8

Combine and simplify the denominator.

Multiply $\frac{\sqrt{3}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{1}{n} \cdot \frac{\sqrt{3} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{1}{n} \cdot \frac{\sqrt{3} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{1}{n} \cdot \frac{\sqrt{3} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

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

$\frac{1}{n} \cdot \frac{\sqrt{3} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{1}{n} \cdot \frac{\sqrt{3} \sqrt{5}}{{\sqrt{5}}^{2}}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{1}{n} \cdot \frac{\sqrt{3} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

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

$\frac{1}{n} \cdot \frac{\sqrt{3} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

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

$\frac{1}{n} \cdot \frac{\sqrt{3} \sqrt{5}}{5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{1}{n} \cdot \frac{\sqrt{3} \sqrt{5}}{5^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{1}{n} \cdot \frac{\sqrt{3} \sqrt{5}}{5^{1}}$

Evaluate the exponent.

$\frac{1}{n} \cdot \frac{\sqrt{3} \sqrt{5}}{5}$

Step 9

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{1}{n} \cdot \frac{\sqrt{3 \cdot 5}}{5}$

Multiply $3$ by $5$ .

$\frac{1}{n} \cdot \frac{\sqrt{15}}{5}$

Step 10

Combine fractions.

Multiply $\frac{1}{n}$ by $\frac{\sqrt{15}}{5}$ .

$\frac{\sqrt{15}}{n \cdot 5}$

Move $5$ to the left of $n$ .

$\frac{\sqrt{15}}{5 n}$

Simplify (2 square root of 3n)/(2 square root of 5n^3)

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