Powers Get Sloppy With Complex Numbers

The fourth root of 1 has many values.

You could do the same without complex numbers:

Let $a=-1$ , then $a^2=1$ . Taking its square root gives $a=1$ .

The error is easy to spot: $a^n=1 \Leftarrow a=1$ always, but $a^n=1 \Rightarrow a=1$ only for real numbers and $n$ odd.

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Note: $\sqrt{b}$ for $b$ a non-negative real number is the real positive number $a$ such that $a^2=b$ and it's well defined because it is unique. But you cannot do the same with $b$ negative or complex. So you should not write $\sqrt{-1}$ . That leads to errors such as $\sqrt{-1}=\sqrt{\frac{1}{-1}} \Rightarrow \sqrt{-1}=\frac{1}{\sqrt{-1}} \Rightarrow -1=1$ . You should prefer to define $i$ as a root of $-1$ , that is, a number such that $i^2=-1$ .

Substituting $i=\sqrt{-1}$ in $i^4=1$ , then

$(\sqrt{-1})^4=1 \ , \ \sqrt{-1}= \sqrt[4]{1}$

$= \sqrt{-1} = 1$

$= -1 = 1^2$

$= -1 = 1$

Thus, it would be invalid.