Simplify ( fourth root of r^20s^9)/( fourth root of r^3s)

$\frac{\sqrt[4]{r^{20} s^{9}}}{\sqrt[4]{r^{3} s}}$

Step 1

Combine $\sqrt[4]{r^{20} s^{9}}$ and $\sqrt[4]{r^{3} s}$ into a single radical.

$\sqrt[4]{\frac{r^{20} s^{9}}{r^{3} s}}$

Step 2

Reduce the expression $\frac{r^{20} s^{9}}{r^{3} s}$ by cancelling the common factors.

Factor $r^{3}$ out of $r^{20} s^{9}$ .

$\sqrt[4]{\frac{r^{3} (r^{17} s^{9})}{r^{3} s}}$

Factor $r^{3}$ out of $r^{3} s$ .

$\sqrt[4]{\frac{r^{3} (r^{17} s^{9})}{r^{3} (s)}}$

Cancel the common factor.

$\sqrt[4]{\frac{r^{3} (r^{17} s^{9})}{r^{3} s}}$

Rewrite the expression.

$\sqrt[4]{\frac{r^{17} s^{9}}{s}}$

Step 3

Cancel the common factor of $s^{9}$ and $s$ .

Factor $s$ out of $r^{17} s^{9}$ .

$\sqrt[4]{\frac{s (r^{17} s^{8})}{s}}$

Cancel the common factors.

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

$\sqrt[4]{\frac{s (r^{17} s^{8})}{s^{1}}}$

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

$\sqrt[4]{\frac{s (r^{17} s^{8})}{s \cdot 1}}$

Cancel the common factor.

$\sqrt[4]{\frac{s (r^{17} s^{8})}{s \cdot 1}}$

Rewrite the expression.

$\sqrt[4]{\frac{r^{17} s^{8}}{1}}$

Divide $r^{17} s^{8}$ by $1$ .

$\sqrt[4]{r^{17} s^{8}}$

Step 4

Rewrite $r^{17} s^{8}$ as ${(r^{4} s^{2})}^{4} r$ .

Factor out $r^{16}$ .

$\sqrt[4]{r^{16} r s^{8}}$

Rewrite $r^{16}$ as ${(r^{4})}^{4}$ .

$\sqrt[4]{{(r^{4})}^{4} r s^{8}}$

Rewrite $s^{8}$ as ${(s^{2})}^{4}$ .

$\sqrt[4]{{(r^{4})}^{4} r {(s^{2})}^{4}}$

Move $r$ .

$\sqrt[4]{{(r^{4})}^{4} {(s^{2})}^{4} r}$

Rewrite ${(r^{4})}^{4} {(s^{2})}^{4}$ as ${(r^{4} s^{2})}^{4}$ .

$\sqrt[4]{{(r^{4} s^{2})}^{4} r}$

Step 5

Pull terms out from under the radical.

$r^{4} s^{2} \sqrt[4]{r}$