Evaluate fourth root of (16x^11y^8)/(81x^7y^6)

$\sqrt[4]{\frac{16 x^{11} y^{8}}{81 x^{7} y^{6}}}$

Step 1

Reduce the expression $\frac{16 x^{11} y^{8}}{81 x^{7} y^{6}}$ by cancelling the common factors.

Factor $x^{7}$ out of $16 x^{11} y^{8}$ .

$\sqrt[4]{\frac{x^{7} (16 x^{4} y^{8})}{81 x^{7} y^{6}}}$

Factor $x^{7}$ out of $81 x^{7} y^{6}$ .

$\sqrt[4]{\frac{x^{7} (16 x^{4} y^{8})}{x^{7} (81 y^{6})}}$

Cancel the common factor.

$\sqrt[4]{\frac{x^{7} (16 x^{4} y^{8})}{x^{7} (81 y^{6})}}$

Rewrite the expression.

$\sqrt[4]{\frac{16 x^{4} y^{8}}{81 y^{6}}}$

Step 2

Cancel the common factor of $y^{8}$ and $y^{6}$ .

Factor $y^{6}$ out of $16 x^{4} y^{8}$ .

$\sqrt[4]{\frac{y^{6} (16 x^{4} y^{2})}{81 y^{6}}}$

Cancel the common factors.

Factor $y^{6}$ out of $81 y^{6}$ .

$\sqrt[4]{\frac{y^{6} (16 x^{4} y^{2})}{y^{6} \cdot 81}}$

Cancel the common factor.

$\sqrt[4]{\frac{y^{6} (16 x^{4} y^{2})}{y^{6} \cdot 81}}$

Rewrite the expression.

$\sqrt[4]{\frac{16 x^{4} y^{2}}{81}}$

Step 3

Rewrite $\frac{16 x^{4} y^{2}}{81}$ as ${(\frac{2 x}{3})}^{4} y^{2}$ .

Factor the perfect power ${(2 x)}^{4}$ out of $16 x^{4} y^{2}$ .

$\sqrt[4]{\frac{{(2 x)}^{4} y^{2}}{81}}$

Factor the perfect power $3^{4}$ out of $81$ .

$\sqrt[4]{\frac{{(2 x)}^{4} y^{2}}{3^{4} \cdot 1}}$

Rearrange the fraction $\frac{{(2 x)}^{4} y^{2}}{3^{4} \cdot 1}$ .

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

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite $\sqrt[4]{y^{2}}$ as $\sqrt{\sqrt{y^{2}}}$ .

$\frac{2 x}{3} \sqrt{\sqrt{y^{2}}}$

Step 6

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

$\frac{2 x}{3} \sqrt{y}$

Step 7

Combine $\frac{2 x}{3}$ and $\sqrt{y}$ .

$\frac{2 x \sqrt{y}}{3}$