As a result of the discussion held on the exercise discussion board about this paradox I'm completely rewriting my previously posted solution.

PrefaceBefore facing the solution of the paradox itself I think it may be worth to make a brief clarification about the exercise wording or presentation. That is:

therefore

As I stated in my previous paradox solution proposal, I think that this formulation can be misunderstood. In the way that the first equation don't necessarily involve the first equality of the second expression.

The initial equation can be understood as

While the second one could be generalized as

I think it's quite obvious that

even if the are equal for

.

Main solutionNonetheless the paradox continues to exist if we consider that

can have two different solutions depending on whether we solve the inner parenthesis first or we multiply the exponents.

Let's resolve this paradox then.

We know that

or that

. So let's apply logarithms to the original expression.

Then,

, where

,

and

are integers.

For simplicity we redefine

and

to get:

Separating the real and imaginary parts:

Therefore, the original equation is only valid if either

or

are zero (or both), and

is equal to its sum.

The function

may be multivalued, and multiple values of

can give the same result for

, but there is only one possible set of parameters

,

and

(and consequently possible values of

) that give the right solution for the original expression. We can't expect to chose

and think we will find a valid one.

I hope this helps.