Question 72·Hard·Advanced Math

Choose $r=1$ and $s=2$. Both are nonzero, as required. Type $r=1$, $s=2$, and the original expression into Desmos. It prints $1.75$. Any equivalent choice must give $1.75$ at this same input.

Type the first three complete choices on separate Desmos lines. They print $0.5$, $3.5$, and $-10$, respectively. None matches the original's $1.75$, so each is ruled out. A choice has to work at every allowed input, including this one.

Type $\frac{2r(2s-3r)(2s+3r)}{s^3}$. Desmos prints $1.75$, matching the original. The other three choices have failed at an allowed input, so this is the only choice left. The factoring approach below shows why it works for every allowed $r$ and $s$. Choice D.

The least common denominator is $s^3$: the first fraction's denominator, $s$, fits into it. Because $s\ne0$, multiply the first fraction's top and bottom by $s^2$:

Subtract the tops now that the bottoms match: $\frac{8rs^2}{s^3}-\frac{18r^3}{s^3}=\frac{8rs^2-18r^3}{s^3}$. Don't subtract the denominators.

Both terms on top contain $2r$, their greatest common factor. Factor it out of both terms:

A difference of squares follows $A^2-B^2=(A-B)(A+B)$. Here the two squares are $(2s)^2$ and $(3r)^2$. Write them that way first:

Apply the pattern, keeping $s^3$ in the denominator:

Since $s\ne0$, these rewrites give the equivalent expression for every allowed input. Choice D.