True/False: If $(a+bi)^3 = 8$, then $a^2+b^2=4$ (yes it's missing the i in b^2)
I believe this is true because if take the cube root of both sides you get $a+bi=2$, squaring both sides will give you $a^2+b^2=2^2=4$
We also know that $a^2+b^2=z*\bar{z}=(a+bi)(a-bi)=a^2-(b^2*i^2) = a^2 - (b^2*(-1))= a^2+b^2$
True/False: If $\operatorname{Arg}(z)=\frac{3\pi}{4}$ and $\operatorname{Arg}(w)=\frac{-\pi}{2}$ then $\operatorname{Arg}(\frac{z}{w}) = \frac{5\pi}{4}$
I believe this is true because when dividing two complex numbers, you divide two complex numbers together you divide their magnitudes and subtract their angles. $\frac{3\pi}{4}-(-\frac{\pi}{2})=\frac{5\pi}{4}$
The reason why I'm asking is because my solution sheet is telling me these are both false but not why, and I think it's wrong.
If you are told that $a$ and $b$ are real numbers then it is true that $a^{2}+b^{2}=4$. Perhaps you are not supposed to assume that these are real.
For the second question the answer depends on how arguments are defined. If you define argument as a number in $[-\pi, \pi)$ then we cannot have a complex number with argument $\frac {5\pi} 4$.