Paradox involving Euler's identity

Question

There is something I fail to understand involving Leonhard Euler's identity:

It is well known that $(e^{2π})^i = 1$.

That means $\sqrt[i]{1} = e^{2π} ≈ 535.49 $.

But there's a rule that states $ \sqrt[a]{1} = 1$ , that means $\sqrt[i]{1} = 1$.

Applying this rule here: if $ a = b, a = c $ it means $ b = c $.

That means $ 535.49 = 1 $?

Answer 1

The only way to define $1^i$ is $=e^{i\ln 1}$ but $\ln 1$ does not have a single value over the complex numbers, so $$\ln 1=0, \pm 2\pi i, \pm 4\pi i, \ldots$$ thus $1^i$ is not a single value, but a set of possible values.

Answer 2

The rule

$$(a^p)^q=a^{pq}$$ doesn't hold for complex exponents, nor

$$(a^p)^{1/p}=a.$$

You just found a counterexample.