After you simplify the numerator, write the whole fraction with the new numerator over $8p^{-1/2}q^3$, and then simplify the numerical part $16/8$ before working with the variables.

When you divide powers with the same base, use $x^m/x^n = x^{m-n}$. Apply this separately to the $p$ terms and the $q$ terms.

If you end up with negative exponents, remember you can move the factor to the denominator (or numerator) to make the exponent positive.

In Desmos, type the original expression exactly (using p and q as variables):

(4*p^(3/2)*q^(-1))^2 / (8*p^(-1/2)*q^3)

This will be your reference expression.

Add p = 2 and q = 3 (or any nonzero values) as separate lines; Desmos will make sliders. These values will be used to compare the expressions numerically.

On new lines, enter each option, for example:

• 2*p^(7/2)/q
• p^3/q^2
• p^4/(2*q^5)
• 2*p^(7/2)/q^5

Desmos will compute a numeric value for each expression using the same $p$ and $q$ as the original.

Look at the numeric value of the original expression and each answer choice. The correct choice is the one whose value matches the original expression for your chosen $p$ and $q$; you can change the slider values (still nonzero) to confirm they continue to match.

Start with the expression

Square each part inside the parentheses:

• $4^2 = 16$
• $(p^{3/2})^2 = p^{3}$ (multiply exponents: $3/2 \cdot 2 = 3$)
• $(q^{-1})^2 = q^{-2}$

So the numerator becomes $16p^3q^{-2}$, and the whole expression is

Now simplify the numbers (coefficients):

So the expression becomes

Use the rule $x^m / x^n = x^{m-n}$ for like bases.

For $p$:

For $q$:

So the expression becomes

A negative exponent means the factor belongs in the denominator: $x^{-n} = 1/x^n$.

So

This matches answer choice D, $\dfrac{2p^{7/2}}{q^{5}}$.