simplifying expressions - How can I get $(r^{1/x})^x$ to simplify to $r$?

Simplify and FullSimplify do not simplify this kind of expression: $(r^\frac{1}{x})^x$. Consider

FullSimplify[r^2+(r^(2/x))^x+(r^(2/(x+y)))^(x+y)]

Mathematica's output is the same as my input, instead of 3 r^2. Is there a way to simplify $(r^\frac{1}{x})^x$ to $r$?

Answer

To expand on rm -rf's hint, the variables need to be positive and real in order for the result you wish to see to hold. You can tell Mathematica to do this using the Assumptions option:

FullSimplify[r^2 + (r^(2/x))^x + (r^(2/(x + y)))^(x + y), 
             Assumptions -> {x > 0, y > 0, r > 0}]
3 r^2

To see that this is really needed, consider an example where it is violated:

r^2 + (r^(2/x))^x + (r^(2/(x + y)))^(x + y) /. {r -> -1, x -> -1/3, y -> -1}
2 - (-1)^(1/3)

which is not equal to 3 r^2. (Thanks to rm -rf for the improved example).

PowerExpand automatically assumes that the variables are real and positive, so you do not need to state it explicitly.