Simplify $\left(\frac{x^3}{27y^6}\right)^{\frac{4}{3}}$.

1. Apply the power to the numerator and denominator.

$\left(\frac{x^3}{27y^6}\right)^{\frac43}=\frac{(x^3)^{\frac43}}{27^{\frac43}(y^6)^{\frac43}}$

2. Simplify each factor using index laws.

$(x^3)^{\frac43}=x^{3\times\frac43}=x^4$
$27^{\frac43}=(\sqrt[3]{27})^4=3^4=81$
$(y^6)^{\frac43}=y^{6\times\frac43}=y^8$
Hence,
$\left(\frac{x^3}{27y^6}\right)^{\frac43}=\frac{x^4}{81y^8}.$