Evaluating fractional exponents: advanced

A common advanced question gives you two variables tied together by a fractional power and asks for their product.
The trick is that the same exponent on two bases combines into one: $a^{1/n} b^{1/n} = (ab)^{1/n}$.

Once the two bases are combined, you have a single power of the product, which you undo by raising both sides to the reciprocal exponent.
For example, if $a^{1/3} b^{1/3} = 3$, then $(ab)^{1/3} = 3$.
Cubing both sides gives $ab = 3^3 = 27$, so the whole product comes out in one step.

A leading coefficient is just cleared first.
Suppose $\frac{1}{3} x^{3/4} y^{3/4} = 1$; multiply both sides by $3$ to get $x^{3/4} y^{3/4} = 3$, which is $(xy)^{3/4} = 3$.
Raising both sides to the reciprocal power $\frac{4}{3}$ gives $xy = 3^{4/3} = 3\sqrt[3]{3}$.

So the plan is always the same three moves.
First combine the like fractional powers into one power of the product; then, if there is a coefficient, divide it out; finally raise both sides to the reciprocal exponent to free the product.
Recognising $a^{1/n} b^{1/n}$ as $(ab)^{1/n}$ is what makes these questions quick.

Find $ab$ if $a^{1/2} b^{1/2} = 5$.
Combine the like powers: $a^{1/2} b^{1/2} = (ab)^{1/2}$, so $(ab)^{1/2} = 5$.
Squaring both sides gives $ab = 5^2 = 25$.

Find $xy$ if $\frac{1}{2} x^{3/4} y^{3/4} = 4$.
Multiply both sides by $2$: $x^{3/4} y^{3/4} = 8$, which is $(xy)^{3/4} = 8$.
Raising both sides to the power $\frac{4}{3}$ gives $xy = 8^{4/3} = (8^{1/3})^4 = 2^4 = 16$.

Find $ab$ if $a^{2/3} b^{2/3} = 4$.
Combine the like powers: $(ab)^{2/3} = 4$.
Raising both sides to the power $\frac{3}{2}$ gives $ab = 4^{3/2} = (4^{1/2})^3 = 2^3 = 8$.