having problem with roots of $x^4=16^{-1}$

Question

The question is $x^4 = \frac{1}{16}, x\in\mathbb C$.

So, after solving this, my answer was different than my other classmates.

Let's just check the first root:

Person 1

$x_0 = (\frac{1}{16})^\frac{1}{4}e^{\frac{π}{8}i}$

Person 2

$x_0 = 1^{\frac{1}{4}}e^{\frac{π}{4}i}$

Me

$x_0 = (\frac{1}{16})^\frac{1}{4}e^{i0}$

So, which answer is the right one?

Answer 1

The equation is a quartic, which means that there are 4 roots in the complex numbers. In fact, the 4 roots are \begin{align}\frac 12,\;\frac i2,\;-\frac12,\;-\frac i2\end{align}

In general, for equations of the form $$x^n=re^{i\theta}$$The $n$ roots are (for $k$ ranging from $0$ to $n-1$)$$\alpha\cdot e^{\frac in\cdot(2\pi k+\theta)}$$where $\alpha$ is the positive real $n$th root of $r$.

Answer 2

We have that

$$x^4 = \frac{1}{16}=\frac{1}{16}e^{i2k\pi} \implies x_k=\frac12e^{ik\frac{\pi}2}$$

with $k=0,1,2,3$ and the principal root is $x_1=\frac12e^{i\frac{\pi}2}=\frac i 2$.