Explanation: Basically consider the modulus and argument separately.
- Modulus: Since a+bi has modulus greater than 1, i.e. |a+bi| > 1, we also know that |a+bi|
1/4 > 1, therefore they should lie outside the unit circle. However, the fourth root of a number greater than 1 will also be
smaller than the original number. Therefore, the fourth roots would also be closer to the origin, than a+bi itself is.
- Argument: In general, if all values for \(\arg(a+bi)\) take the form \(\arg(a+bi) = \theta + 2k\pi\) for some integer \(k\), then all values for \( \arg((a+bi)^{1/4})\) take the form \( \arg((a+bi)^{1/4}) = \frac{\theta}{4} + \frac\pi2 k\). This can be proven using the standard algebraic method of explicitly computing fourth roots.
Therefore, we know that the first value for \( \arg((a+bi)^{1/4})\) will be one quarter of the argument of a+bi, which gives the first root. After that, we see that the other values require us to add \( \frac\pi2\) to the argument to obtain all of the other roots.