When you divide powers with the same base (like $x^{1/3}$ over $x^{-2/3}$), subtract the exponents: numerator exponent minus denominator exponent. Do this for both $x$ and $y$.

After using the exponent rules, you will get a negative exponent on $y$. Remember that $a^{-n} = \dfrac{1}{a^n}$, so rewrite with positive exponents in the denominator.

Once the inside is simplified to a single fraction, apply the exponent of 2 to both the numerator and denominator. Be careful to square both the number and the variable parts.

In Desmos, choose convenient nonzero values for $x$ and $y$ (for example, create sliders $a$ and $b$ and use them in place of $x$ and $y$). Type the original expression as something like $((2a^{1/3}b^{-2})/(4a^{-2/3}b))^2$ and note its numerical value for a particular pair $(a,b)$.

Enter each answer choice as a separate expression, replacing $x$ with $a$ and $y$ with $b$ (for example, one line as $a^2/(4b^6)$, another as $4a^2b^{-6}$, etc.). For several different values of $a$ and $b$, see which choice always has the same numerical value as the original expression; that choice is the equivalent expression.

Start with the expression

First, simplify the coefficient (number) part of the fraction inside the parentheses:

• The numbers are $2$ in the numerator and $4$ in the denominator, so $\dfrac{2}{4} = \dfrac{1}{2}$.

So the expression inside becomes

Apply the rule for dividing powers with the same base: when you divide, you subtract exponents (numerator exponent minus denominator exponent).

For $x$:

For $y$:

Now the whole expression inside the parentheses becomes

The factor $y^{-3}$ means $\dfrac{1}{y^3}$. So

Now the original expression is

When you square a fraction, you square both the numerator and the denominator:

Now square the denominator:

• $2^2 = 4$
• $(y^3)^2 = y^{3\cdot 2} = y^6$

So the expression becomes

This matches choice C.