Where is the mistake? $-1=(-1)^{2/2}=\left(\left (-1\right)\displaystyle ^2\right)^{1/2}=1^{1/2}=\sqrt 1=1$

Question

Where is the mistake? $$-1=(-1)^{2/2}=\left(\left (-1\right)\displaystyle ^2\right)^{1/2}=1^{1/2}=\sqrt 1=1$$

Answer 1

Each $=$ but the second is clearly right. That the second $=$ is wrong proves $\color{blue}{x^{a/b}=(x^a)^{1/b}}$ admits counterexamples with $x<0$. We use $x>0$ in the manipulation$$(x^{a/b})^b=x^a\implies\color{blue}{x^{a/b}=(x^a)^{1/b}},$$which is our only reason to expect the blue part to be true.