I was just bored and started practicing even more the exponentiations, and as I was working, I went ahead and did this:
$(-1)^{2} = ((-1)^{\sqrt{2}})^{\sqrt{2}}$
So, I entered in my calculator of what is negative one to the power of the square root of two and it gave me negative one, so it goes like:
$(-1)^{2} = ((-1)^{\sqrt{2}})^{\sqrt{2}} = (-1)^{\sqrt{2}} = (-1)$
But in other hand, we have:
$(-1)^2 = (-1)(-1) = 1$
And that implies that $-1 = +1$... Is there something wrong in it? If not, could someone explain me how can that happen?
Thanks for the answers.
Here you have $2$,$3$ mistakes:
$(-1)^{2} = ((-1)^{\sqrt{2}})^{\sqrt{2}} = (-1)^{\sqrt{2}} = (-1)$
The first equality is true, since $\sqrt 2\cdot \sqrt 2=2$. The expression $(-1)^{\sqrt{2}}$ is not real and certainly not $-1$.