Simplify ( square root of 63mn)/( square root of 252mn^3)

$\frac{\sqrt{63 m n}}{\sqrt{252 m n^{3}}}$

Step 1

Combine $\sqrt{63 m n}$ and $\sqrt{252 m n^{3}}$ into a single radical.

$\sqrt{\frac{63 m n}{252 m n^{3}}}$

Step 2

Reduce the expression $\frac{63 m n}{252 m n^{3}}$ by cancelling the common factors.

Factor $63$ out of $63 m n$ .

$\sqrt{\frac{63 (m n)}{252 m n^{3}}}$

Factor $63$ out of $252 m n^{3}$ .

$\sqrt{\frac{63 (m n)}{63 (4 m n^{3})}}$

Cancel the common factor.

$\sqrt{\frac{63 (m n)}{63 (4 m n^{3})}}$

Rewrite the expression.

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

Step 3

Cancel the common factor of $m$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Cancel the common factor of $n$ and $n^{3}$ .

Raise $n$ to the power of $1$ .

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

Factor $n$ out of $n^{1}$ .

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

Cancel the common factors.

Factor $n$ out of $4 n^{3}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Rewrite $1$ as $1^{2}$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{2 n}$