Our A-Level textbook gives an example where given complex numbers $z_1,z_2$, it was proven that $|z_1z_2|=|z_1||z_2|$ and that $arg(z_1z_2) = arg(z_1) + arg(z_2)$. This is done by considering the complex numbers in the form of $r(cos\theta+isin\theta)$
$z_1z_2=r_1(cos\theta_1+isin\theta_1)r_2(cos\theta_2+isin\theta_2)$ which when expanded and invoking the addition formulae for sin and cos yields
$r_1r_2[cos(\theta_1+\theta_2)+i(sin(\theta_1+\theta_2)]$
This completes the proof.
It then continues by saying if $z_1$ and $z_2$ are written in exponential form $r_1e^{i\theta_1}$ and $r_2e^{i\theta_2}$ respectively, based on the results from above,
$z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}$
It then displays a Watch Out message that says
"You cannot automatically assume the law of indices work the same way with complex numbers as with real numbers. This results only shows that they can be applied in these specific cases"
I am confused by the last sentence. Do they mean that $z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}$ is fortuitous? That it is not a result of applying the law of indices?
I guess my question is, is it true, for $\theta, \beta\;real $, invoking the indices law for real numbers that
$e^{i\theta}\times e^{i\beta}=e^{i(\theta+\beta)}?$