Let p be a prime for which the expression is a perfect power.

p cannot be 2, since the expression then equals 13, which is not a perfect power.

So p is odd, and so we can write:

2^{p} + 3^{p} = (2 + 3)(2^{p-1} - 2^{p-2}3 + 2^{p-3}3^{2} - ... + 3^{p-1})

The expression is therefore a multiple of 5.
Since it is a perfect power, 5^{2} must divide the expression,
and so 5 must divide evenly into the second factor on the right hand side.

But modulus 5, we have:

2^{p-1} - 2^{p-2}3 + 2^{p-3}3^{2} - ... + 3^{p-1}

= 2^{p-1} - 2^{p-2}(-2) + 2^{p-3}(-2)^{2} - ... + (-2)^{p-1} (mod 5)

= 2^{p-1} + 2^{p-1} + ... + 2^{p-1} (mod 5)

= p2^{p-1} (mod 5)

Now 5 is not a factor of 2^{p-1}, so 5 must divide evenly into p.
Since p is prime, we conclude that the only case where the expression can be a prime power
with p odd is p = 5.
But for that case, the expression is equal 32 + 243 = 275, which is not a perfect power.

Hence 2^{p} + 3^{p} is never a perfect power for any prime p.