Recently I had a thought that got me thinking.
There is the identity $\left (a^b\right )^c=a^{bc}$.
If I now take this identity I can do this:$$(-1)=(-1)^{2\cdot 0.5}=\left ((-1)^2\right )^{0.5}=1^{0.5}=1.$$Isn't an identity supposed to be true for all values?
Interestingly if I take the part $(-1)^{0.5\cdot 2}$ I can rearange it to both:$$\left ((-1)^2\right )^{0.5}=1^{0.5}=1$$and$$\left ((-1)^{0.5}\right )^2=i^2=-1.$$So in the latter example the identity does hold true.
But isn't exponentiation supposed to be commutative? Why do I get two different solutions?
Edit: As a user has pointed out in Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$? The identity a^0.5b^0.5=(ab)^0.5 only holds true for postive numbers.
Is that also the case for the identity I used? Are all identities covering exponents only true for positive numbers?
Also what I meant with commutative was: (a^b)^c=(a^c)^b?