I want the distance from $1$ to $-2 i^\frac 2 3 = -2(-1)^\frac 1 3$.
So I run in Mathematica Norm[1 + 2 * i^(2/3)] which gives me sqrt(7).
I totally don't understand the background, and I have beginner knowledge on complex numbers. I use this in differential equations problems.
So, this is not equal with the following?
$\sqrt{|1^2+2^2(-1)^\frac 2 3|}=\sqrt{|1+4\cdot 1^\frac 1 3|} = \sqrt 5$ ?
Write $z = 1 + 2(-1)^{\frac{1}{3}} = 1 + 2 (e^{i\pi})^{\frac{1}{3}} = 1 + 2e^{\frac{\pi}{3}i} = 1 + 2[\cos \frac{\pi}{3} + i \sin \frac{\pi}{3} ] = 2 + i\sqrt{3}$.
$|z|^{2} = z \cdot \overline{z} = 4 + 3 = 7$.