Can an irrational power of irrational number be a rational number?

Question

anonymous · Answer

yes

asnaseer · Answer

$$e^{i\pi}=-1$$

anonymous · Answer

is this a quiz? if so i will be quiet
@asneer that is not an example since $i\pi$ is not real

blockcolder · Answer

$$(\sqrt2^{\sqrt{2}})^{\sqrt{2}}=2$$
Since $\sqrt2^{\sqrt2}$ is irrational, we are done.

asnaseer · Answer

nothing in the question said it had to be "real"

anonymous · Answer

idea is to take $\sqrt{2}^{\sqrt{2}}$

anonymous · Answer

it said "irrational" a condition that only applies to reals right?

asnaseer · Answer

really - I never knew that

asnaseer · Answer

so irrational numbers don't exist in the complex plane?

anonymous · Answer

@blockholder not quite, since you have not shown $\sqrt{2}^{\sqrt{2}}$ is irrational, so that is close, but not quite right

blockcolder · Answer

Well, it is either rational or not. If rational, we have our answer. Otherwise, raise it to $\sqrt{2}$ and we're done.

anonymous · Answer

rational: any real number that can be written as $\frac{a}{b}$ where a, b are integers.  that definition doesn't apply to complex numbers. we don't say for example that $2i$ is rational whereas $\sqrt{2}i$ is not

anonymous · Answer

now it is correct!

asnaseer · Answer

@satellite73 - you are right http://en.wikipedia.org/wiki/Irrational_number good - I learnt something new today :)

anonymous · Answer

;)