If r>0 and (9r/2)^(1/3) = (1/2) r, what is the value of r?

Two good ways to go here. First, the algebra:

One possible solution to the above is *r* = 0, but the question says that *r* > 0, so we can ignore that and move on:

We need not consider *r* = –6 because we know *r* is positive.

The *other* way to go is to graph both sides of the equation and find the intersection(s).

Because *r* is positive (my software made me use *x* in the graph above, as your calculator probably will, so just remember that *x* = *r*), the only intersection that matters is (6, 3). At that intersection, *x* = 6, so we know that the value of *r* is 6.