Question 69·Medium·Advanced Math

The $\frac23$ is outside the parentheses, so it applies to both factors inside: $(x^{3/2}y^{-6})^{2/3}=(x^{3/2})^{2/3}(y^{-6})^{2/3}$.

For a power of a power, multiply the inside exponent by the outside exponent, rather than adding them: $(x^{3/2})^{2/3}(y^{-6})^{2/3}=x^{(3/2)(2/3)}y^{(-6)(2/3)}$.

Evaluate those products, keeping the minus sign on $y$: $x^{(3/2)(2/3)}y^{(-6)(2/3)}=x^1y^{-4}$. The exponent on $x$ is $1$, while the exponent on $y$ is $-4$.

Pick $x_1=8$ and $y_1=2$, which satisfy the given conditions and won't hide an exponent slip the way $1$ could. Type the original expression and $x_1y_1^{-4}$ on separate lines. Desmos prints $0.5$ for each. One matching input is a check, not proof; the exponent rules above give the exact match.

A negative exponent means a reciprocal: $y^{-4}=\frac1{y^4}$. This is defined because $y\neq0$. Rewrite the result: $x^1y^{-4}=\frac{x}{y^4}$. So the equivalent expression is $\frac{x}{y^4}$. Choice D.