I'm currently going through Introduction to Classical and Quantum Computing, by Thomas Wong, and I'm struggling with exercise 2.8 (page 86):
Exercise 2.8. A qubit is in the state $$ \frac{e^{i\pi/8}}{\sqrt{5}}\left| 0 \right> + \beta\left| 1 \right> $$ What is a possible value of $\beta$ ?
The answer given for the exercise at the back of the textbook (page 358) is $$2/\sqrt{5}e^{i\theta}$$. Though, when doing the math I arrived at a slightly different answer:
$$\begin{align} \left( \frac{e^{i\pi/8}}{\sqrt{5}} \right) \left( \frac{e^{-i\pi/8}}{\sqrt{5}} \right) + \beta^2 &=1\\ \frac{e^0}{5} + \beta^2 &= 1\\ \beta^2&=\frac{4}{5}\\ \beta&=\frac{2}{\sqrt{5}} \end{align}$$
If my understanding is correct, both answers are correct as their total probablity is equal to 1. What I don't understand is how the author arrives at this different answer, and whether or not it's "more valid" than mine.