Simplify the Radical Expression ((-54x^9)/(y^4))^(2/3)

${(\frac{- 54 x^{9}}{y^{4}})}^{\frac{2}{3}}$

Move the negative in front of the fraction.

${(- \frac{(54) x^{9}}{y^{4}})}^{\frac{2}{3}}$

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

Apply the product rule to $- \frac{54 x^{9}}{y^{4}}$ .

${(- 1)}^{\frac{2}{3}} {(\frac{54 x^{9}}{y^{4}})}^{\frac{2}{3}}$

Apply the product rule to $\frac{54 x^{9}}{y^{4}}$ .

${(- 1)}^{\frac{2}{3}} \frac{{(54 x^{9})}^{\frac{2}{3}}}{{(y^{4})}^{\frac{2}{3}}}$

Apply the product rule to $54 x^{9}$ .

${(- 1)}^{\frac{2}{3}} \frac{54^{\frac{2}{3}} {(x^{9})}^{\frac{2}{3}}}{{(y^{4})}^{\frac{2}{3}}}$

Rewrite $- 1$ as ${(- 1)}^{3}$ .

${({(- 1)}^{3})}^{\frac{2}{3}} \frac{54^{\frac{2}{3}} {(x^{9})}^{\frac{2}{3}}}{{(y^{4})}^{\frac{2}{3}}}$

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

${(- 1)}^{3 (\frac{2}{3})} \frac{54^{\frac{2}{3}} {(x^{9})}^{\frac{2}{3}}}{{(y^{4})}^{\frac{2}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(- 1)}^{3 (\frac{2}{3})} \frac{54^{\frac{2}{3}} {(x^{9})}^{\frac{2}{3}}}{{(y^{4})}^{\frac{2}{3}}}$

Rewrite the expression.

${(- 1)}^{2} \frac{54^{\frac{2}{3}} {(x^{9})}^{\frac{2}{3}}}{{(y^{4})}^{\frac{2}{3}}}$

Raise $- 1$ to the power of $2$ .

$1 \frac{54^{\frac{2}{3}} {(x^{9})}^{\frac{2}{3}}}{{(y^{4})}^{\frac{2}{3}}}$

Multiply $\frac{54^{\frac{2}{3}} {(x^{9})}^{\frac{2}{3}}}{{(y^{4})}^{\frac{2}{3}}}$ by $1$ .

$\frac{54^{\frac{2}{3}} {(x^{9})}^{\frac{2}{3}}}{{(y^{4})}^{\frac{2}{3}}}$

Multiply the exponents in ${(x^{9})}^{\frac{2}{3}}$ .

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

$\frac{54^{\frac{2}{3}} x^{9 (\frac{2}{3})}}{{(y^{4})}^{\frac{2}{3}}}$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$\frac{54^{\frac{2}{3}} x^{3 (3) \frac{2}{3}}}{{(y^{4})}^{\frac{2}{3}}}$

Cancel the common factor.

$\frac{54^{\frac{2}{3}} x^{3 \cdot 3 \frac{2}{3}}}{{(y^{4})}^{\frac{2}{3}}}$

Rewrite the expression.

$\frac{54^{\frac{2}{3}} x^{3 \cdot 2}}{{(y^{4})}^{\frac{2}{3}}}$

Multiply $3$ by $2$ .

$\frac{54^{\frac{2}{3}} x^{6}}{{(y^{4})}^{\frac{2}{3}}}$

Multiply the exponents in ${(y^{4})}^{\frac{2}{3}}$ .

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

$\frac{54^{\frac{2}{3}} x^{6}}{y^{4 (\frac{2}{3})}}$

Multiply $4 (\frac{2}{3})$ .

Combine $4$ and $\frac{2}{3}$ .

$\frac{54^{\frac{2}{3}} x^{6}}{y^{\frac{4 \cdot 2}{3}}}$

Multiply $4$ by $2$ .

$\frac{54^{\frac{2}{3}} x^{6}}{y^{\frac{8}{3}}}$

Reorder factors in $\frac{54^{\frac{2}{3}} x^{6}}{y^{\frac{8}{3}}}$ .

$\frac{x^{6} \cdot 54^{\frac{2}{3}}}{y^{\frac{8}{3}}}$