Question 240·Medium·Advanced Math

Type $p=4$, $q=2$, and the original expression into Desmos. It shows $8$. Both inputs keep the expression defined, and $4$ has a simple square root. Use this value to check your work later; one matching input alone would not prove equivalence.

Give the outside $2$ to every factor, including $4$. Multiply each variable’s exponent by $2$:

Keeping $q$’s minus sign matters: squaring $q^{-1}$ gives $q^{-2}$, not $q^2$.

The denominator starts with $8$, so divide the squared coefficient $16$ by $8$:

The variables still need to be divided.

For a quotient of powers with the same base, subtract the bottom exponent. The bottom $p$ exponent is $-\frac12$, so subtract a negative:

Use the same quotient rule for $q$: subtract the bottom exponent $3$ from the top exponent $-2$:

A negative exponent means a reciprocal, so move $q^5$ to the denominator:

Type this form into Desmos. It also shows $8$ for $p=4$ and $q=2$, matching the original; the exponent rules above establish the match for every allowed input. The equivalent expression is $\frac{2p^{7/2}}{q^5}$. Choice D.