Simplify ( square root of 48n^6)/( square root of 6n^3)

$\frac{\sqrt{48 n^{6}}}{\sqrt{6 n^{3}}}$

Step 1

Combine $\sqrt{48 n^{6}}$ and $\sqrt{6 n^{3}}$ into a single radical.

$\sqrt{\frac{48 n^{6}}{6 n^{3}}}$

Step 2

Reduce the expression $\frac{48 n^{6}}{6 n^{3}}$ by cancelling the common factors.

Factor $6$ out of $48 n^{6}$ .

$\sqrt{\frac{6 (8 n^{6})}{6 n^{3}}}$

Factor $6$ out of $6 n^{3}$ .

$\sqrt{\frac{6 (8 n^{6})}{6 (n^{3})}}$

Cancel the common factor.

$\sqrt{\frac{6 (8 n^{6})}{6 n^{3}}}$

Rewrite the expression.

$\sqrt{\frac{8 n^{6}}{n^{3}}}$

Step 3

Cancel the common factor of $n^{6}$ and $n^{3}$ .

Factor $n^{3}$ out of $8 n^{6}$ .

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

Cancel the common factors.

Multiply by $1$ .

$\sqrt{\frac{n^{3} (8 n^{3})}{n^{3} \cdot 1}}$

Cancel the common factor.

$\sqrt{\frac{n^{3} (8 n^{3})}{n^{3} \cdot 1}}$

Rewrite the expression.

$\sqrt{\frac{8 n^{3}}{1}}$

Divide $8 n^{3}$ by $1$ .

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

Step 4

Rewrite $8 n^{3}$ as ${(2 n)}^{2} \cdot (2 n)$ .

Factor $4$ out of $8$ .

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

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

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

Factor out $n^{2}$ .

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

Move $2$ .

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

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

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

Add parentheses.

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

Step 5

Pull terms out from under the radical.

$2 n \sqrt{2 n}$