Few days ago a someone asked me a questions: $2^{-i}$, after answering it I saw that $-i=\frac{1}{i}$ so the question can be rewritten to $\sqrt[i]{2}$, after seeing this i tried to solve for general $z\in\mathbb{C}$ and I got to the next expration: $$\sqrt[i]{z}=\sqrt[i]{re^{\varphi i}}=\left(re^{\varphi i}\right)^{-i}=e^{-i\ln\left(re^{\varphi i}\right)}=e^{-i\ln(r)+\varphi+2k\pi}=e^{\varphi+2k\pi}e^{-i\ln(r)}=e^{\varphi+2k\pi}\left(\cos\left(\ln(r)\right)-i\sin\left(\ln(r)\right)\right)$$
Now it is very confusing for me, why does the Arg of the answer is based on the length of $z$ and the length of the answer is based on the Arg of $z$? Is there a good way to understand this by logic? Or could it be that i made a mistake in the middle?
$$\begin{array}{rcl} z^{-i} &=& \exp(-i\ln z) \\ &=& \exp(-i\ln(re^{i\varphi})) \\ &=& \exp(-i[\ln_\Bbb R (r) + i\varphi + 2ni\pi]) \\ &=& \exp(-i\ln_\Bbb R (r) + \varphi - 2n\pi) \\ &=& \exp(\varphi - 2n\pi) [\cos(\ln r)-i\sin(\ln r)] \\ \end{array}$$
For an understanding, we must view numbers and exponentiation in a different angle. Firstly, exponentiation is not repeated multiplication.
We'll start with the real numbers before going to the complex numbers. We'll think of numbers as their actions to the real line (and to the complex plane when we go to the complex numbers).
A number has two forms: as an adder and as a multiplier. An adder shifts the real line left or right. A multiplier stretches the real line. Adders can combine with adders, and multipliers with multipliers. For example, $3$ as an adder shifts the real line to the right by $3$ units.
Exponentiation transform adders to multipliers. We have a remarkable property of exponentiation: "transforming each adder into multiplier and then combining them as multipliers" is equivalent to "combining the adders and then transform the new adder into a multiplier". They give you the same multiplier. For example, we have the adders $1$ and $2$. We can combine them first and then transform to get $a^{1+2}$, or we can transform them first and then combine to get $a^1 \times a^2$. We get the same result.
We define the base as where it moves $1$ to after it transforms $1$ (multiplying $1$ to $a^1$). Note that we define the base after the exponentiation.
Also, note that the extent of stretching after the transformation depends on the base (if it transforms $1$ to a larger multiplier (a multiplier that stretches more), then it would transform other numbers also to a larger multiplier).
Now we are ready to go to the complex numbers.
We define an adder as a number that keeps the distances between the grid lines unchanged and leave horizontal lines horizontal. We define a multiplier as a number that keeps the origin where it is and bring other points to new points while keeping the relative distance between grid lines unchanged.
We see that these definitions in the real numbers correspond to "shifting" and "stretching" the real line respectively.
However, in the complex numbers, we have an extra dimension. Instead of shifting horizontally, we can also shift vertically. Instead of just stretching, we can satisfy our definition by rotating the complex plane.
As inherited from the transformation of the real line, if the base is real, a horizontal adder is transformed to a stretching multiplier. It would only make sense then, to transform a vertical adder (e.g. $i$) to a rotating multiplier (so $r^i$ is a rotating multiplier where $r$ is a real number). The extent of rotation (the argument of $z^i$) would depend on the magnitude of the base (the magnitude of $z$).
We state again what we just discovered because it is very important: the argument of $z^i$ depends on the magnitude of $z$.
If it transforms $1$ to a purely rotational multiplier, i.e. the base lives on the unit circle, then one would expect it to transform $i$ to a purely stretching multiplier. The extent of the stretch (the magnitude of $z^i$) would depend on the extent of the rotation (the argument of $z$).
We state again what we just discovered because it is very important: the magnitude of $z^i$ depends on the argument of $z$.
The two statements in bold are what you are asking for. I inherited this intuition from 3Blue1Brown. I've also created a Desmos program to let you play with multipliers. It is $[R\exp(iT)]^{a+bi}$.