Confused about complex numbers

Solution 1:

It's exactly as if you would write:

I am confused about something: $$((-1)^2)^{0.5} = (-1)^{2\cdot 0.5} = (-1)^1 = -1$$ but $$(-1)^2=1 \text{ and } 1^{0.5} = 1$$ where is my mistake??

Solution 2:

When dealing with complex numbers, you no longer interchange radicals with fractional exponents, in the same way you did with positive real numbers.

If $x$ is real, $\sqrt[3]x$ denotes the principal cube root of $x$, and has only one value. In this way $8^{1/3}$ and $\sqrt[3]{8}$ mean the same thing.

But if we are doing algebra with complex numbers, and we wrote $5^{1/3}$, we would mean any solution to $z^3 - 5 = 0$.

So $1^{1/2}$ is defined as any solution to $z^2 - 1 = 0$. $z = 1$ and $z = -1$ are both solutions to this.