Simplify ( fourth root of 243x^12y^20)/( fourth root of 3x^4y^16)

$\frac{\sqrt[4]{243 x^{12} y^{20}}}{\sqrt[4]{3 x^{4} y^{16}}}$

Step 1

Combine $\sqrt[4]{243 x^{12} y^{20}}$ and $\sqrt[4]{3 x^{4} y^{16}}$ into a single radical.

$\sqrt[4]{\frac{243 x^{12} y^{20}}{3 x^{4} y^{16}}}$

Step 2

Reduce the expression $\frac{243 x^{12} y^{20}}{3 x^{4} y^{16}}$ by cancelling the common factors.

Factor $3$ out of $243 x^{12} y^{20}$ .

$\sqrt[4]{\frac{3 (81 x^{12} y^{20})}{3 x^{4} y^{16}}}$

Factor $3$ out of $3 x^{4} y^{16}$ .

$\sqrt[4]{\frac{3 (81 x^{12} y^{20})}{3 (x^{4} y^{16})}}$

Cancel the common factor.

$\sqrt[4]{\frac{3 (81 x^{12} y^{20})}{3 (x^{4} y^{16})}}$

Rewrite the expression.

$\sqrt[4]{\frac{81 x^{12} y^{20}}{x^{4} y^{16}}}$

Step 3

Cancel the common factor of $x^{12}$ and $x^{4}$ .

Factor $x^{4}$ out of $81 x^{12} y^{20}$ .

$\sqrt[4]{\frac{x^{4} (81 x^{8} y^{20})}{x^{4} y^{16}}}$

Cancel the common factors.

Factor $x^{4}$ out of $x^{4} y^{16}$ .

$\sqrt[4]{\frac{x^{4} (81 x^{8} y^{20})}{x^{4} (y^{16})}}$

Cancel the common factor.

$\sqrt[4]{\frac{x^{4} (81 x^{8} y^{20})}{x^{4} y^{16}}}$

Rewrite the expression.

$\sqrt[4]{\frac{81 x^{8} y^{20}}{y^{16}}}$

Step 4

Cancel the common factor of $y^{20}$ and $y^{16}$ .

Factor $y^{16}$ out of $81 x^{8} y^{20}$ .

$\sqrt[4]{\frac{y^{16} (81 x^{8} y^{4})}{y^{16}}}$

Cancel the common factors.

Multiply by $1$ .

$\sqrt[4]{\frac{y^{16} (81 x^{8} y^{4})}{y^{16} \cdot 1}}$

Cancel the common factor.

$\sqrt[4]{\frac{y^{16} (81 x^{8} y^{4})}{y^{16} \cdot 1}}$

Rewrite the expression.

$\sqrt[4]{\frac{81 x^{8} y^{4}}{1}}$

Divide $81 x^{8} y^{4}$ by $1$ .

$\sqrt[4]{81 x^{8} y^{4}}$

Step 5

Rewrite $81 x^{8} y^{4}$ as ${(3 x^{2} y)}^{4}$ .

$\sqrt[4]{{(3 x^{2} y)}^{4}}$

Step 6

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

$3 x^{2} y$