what is the difference - sorry for over-simplicity

Question

i am asking too simple question, sorry for that. what is the difference between these two imaginär numbers?

$\operatorname{Im}(| \sqrt2+3i|^2)$ vs. $\operatorname{Im}((\sqrt2+3i)^2)$

$| \sqrt2+3i|^2$ means $ \sqrt2^2+3i^2$ right?

Answer 1

Short answer: the first one is zero since $\vert \sqrt{2}+3i\vert^2$ is a real number (it's the modulus of your complex number).

Long answer: compute them and see! Recall that $\vert x+iy\vert=\sqrt{x^2+y^2}$:

$$ \vert \sqrt 2+3i\vert^2=\sqrt{2}^2+3^2=11\\ (\sqrt{2}+3i)^2=\sqrt{2}^2+(3i)^2+2(\sqrt{2}\cdot 3i)=-7+6\sqrt{2}i $$

So the imaginary part of the second number is $6\sqrt{2}$.