Simplify (-6xy^2)^(1/4)( fourth root of -6xy)^3

${(- 6 x y^{2})}^{\frac{1}{4}} {(\sqrt[4]{- 6 x y})}^{3}$

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- 6 x y^{2}$ .

${(- 6 x)}^{\frac{1}{4}} {(y^{2})}^{\frac{1}{4}} {(\sqrt[4]{- 6 x y})}^{3}$

Apply the product rule to $- 6 x$ .

${(- 6)}^{\frac{1}{4}} x^{\frac{1}{4}} {(y^{2})}^{\frac{1}{4}} {(\sqrt[4]{- 6 x y})}^{3}$

Step 2

Multiply the exponents in ${(y^{2})}^{\frac{1}{4}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(- 6)}^{\frac{1}{4}} x^{\frac{1}{4}} y^{2 (\frac{1}{4})} {(\sqrt[4]{- 6 x y})}^{3}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

${(- 6)}^{\frac{1}{4}} x^{\frac{1}{4}} y^{2 \frac{1}{2 (2)}} {(\sqrt[4]{- 6 x y})}^{3}$

Cancel the common factor.

${(- 6)}^{\frac{1}{4}} x^{\frac{1}{4}} y^{2 \frac{1}{2 \cdot 2}} {(\sqrt[4]{- 6 x y})}^{3}$

Rewrite the expression.

${(- 6)}^{\frac{1}{4}} x^{\frac{1}{4}} y^{\frac{1}{2}} {(\sqrt[4]{- 6 x y})}^{3}$

Step 3

Rewrite ${\sqrt[4]{- 6 x y}}^{3}$ as ${({(- 6 x y)}^{3})}^{\frac{1}{4}}$ .

${(- 6)}^{\frac{1}{4}} x^{\frac{1}{4}} y^{\frac{1}{2}} \sqrt[4]{{(- 6 x y)}^{3}}$

Step 4

Apply the product rule to $- 6 x y$ .

${(- 6)}^{\frac{1}{4}} x^{\frac{1}{4}} y^{\frac{1}{2}} \sqrt[4]{{(- 6 x)}^{3} y^{3}}$

Step 5

Apply the product rule to $- 6 x$ .

${(- 6)}^{\frac{1}{4}} x^{\frac{1}{4}} y^{\frac{1}{2}} \sqrt[4]{{(- 6)}^{3} x^{3} y^{3}}$

Step 6

Raise $- 6$ to the power of $3$ .

${(- 6)}^{\frac{1}{4}} x^{\frac{1}{4}} y^{\frac{1}{2}} \sqrt[4]{- 216 x^{3} y^{3}}$