Focus on the part where one expression is divided by another. For each base ($p$ and $q$), subtract the exponent in the denominator from the exponent in the numerator.

For $(p^{1/2} q^{-4/3})^{-2}$, use the rule $(a^m)^n = a^{mn}$ on each factor separately, then combine the exponents of $p$ and $q$ from all parts of the expression.

Choose convenient positive values for $p$ and $q$, such as $p=4$ and $q=9$. In Desmos, type p = 4 and q = 9 on separate lines so Desmos treats them as constants.

On a new line, type the original expression using exponents: for example, (p^5*q)^(1/3) / sqrt(p*q^3) * (p^(1/2)*q^(-4/3))^(-2). Desmos will show a numeric value for this line.

For each choice A–D, type a new line with its expression exactly as written in the problem (using the same $p$ and $q$). Compare the numeric outputs: the choice whose value matches the original expression for this set of $p$ and $q$ (and, if you like, for a second set such as $p=3$, $q=2$) is the equivalent expression.

Use the rule that the nth root of $x^m$ is $x^{m/n}$. Rewrite the cube root as $(p^5 q)^{1/3} = p^{5/3} q^{1/3}$ and the square root as $(p q^3)^{1/2} = p^{1/2} q^{3/2}$. So the whole expression becomes $p^{5/3} q^{1/3}$ divided by $p^{1/2} q^{3/2}$, multiplied by $(p^{1/2} q^{-4/3})^{-2}$. In other words, it is $(p^{5/3} q^{1/3}) / (p^{1/2} q^{3/2})$ times $(p^{1/2} q^{-4/3})^{-2}$.

Now simplify the fraction $p^{5/3} q^{1/3} / (p^{1/2} q^{3/2})$ by subtracting exponents for like bases (numerator minus denominator). For $p$, compute $5/3 - 1/2$. With common denominator 6, this is $10/6 - 3/6 = 7/6$, so the exponent on $p$ is $7/6$. For $q$, compute $1/3 - 3/2$. With denominator 6, this is $2/6 - 9/6 = -7/6$, so the exponent on $q$ is $-7/6$. After this step, the expression is $p^{7/6} q^{-7/6} (p^{1/2} q^{-4/3})^{-2}$.

Use the rule $(a^m)^n = a^{mn}$ on $(p^{1/2} q^{-4/3})^{-2}$. For $p$, multiply exponents: $(1/2) \cdot (-2) = -1$, so you get $p^{-1}$. For $q$, multiply exponents: $(-4/3) \cdot (-2) = 8/3$, so you get $q^{8/3}$. So $(p^{1/2} q^{-4/3})^{-2}$ simplifies to $p^{-1} q^{8/3}$, and the whole expression is now $p^{7/6} q^{-7/6} p^{-1} q^{8/3}$.

Combine the $p$ terms and the $q$ terms by adding their exponents. For $p$, add $7/6$ and $-1$: $7/6 - 1 = 7/6 - 6/6 = 1/6$, so the overall exponent on $p$ is $1/6$. For $q$, add $-7/6$ and $8/3$. Rewrite $8/3$ as $16/6$, so $-7/6 + 16/6 = 9/6 = 3/2$, so the overall exponent on $q$ is $3/2$. Therefore, the entire expression simplifies to $p^{1/6} q^{3/2}$, which matches answer choice A.