# Finding the nth root of a complex number?

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1. Mar 14, 2015

### 21joanna12

1. The problem statement, all variables and given/known data
Find the solutions to $z^{\frac{3}{4}}=\sqrt{6}+\sqrt{2}i$

2. Relevant equations
de Moivre's theorem

3. The attempt at a solution
$z^{\frac{3}{4}}=2\sqrt{2}e^{\frac{\pi i}{6}}=2\sqrt{2}e^{\frac{\pi i}{6}+2k\pi}=2\sqrt{2}e^{\frac{\pi +12k\pi}{6}i}$

$z=4e^{\frac{4}{3}{\frac{\pi +12k\pi}{6}i}=4e^{\frac{2\pi +24k\pi}{9}i}$

For answers with a principal argument,

$-\pi<\frac{2\pi +24k\pi}{9}\leq\pi$
$-\frac{11}{24}<k\leq\frac{7}{24}$

So the only integer value of k satisfying this is zero. So I get $z=4e^{\frac{2\pi}{9}i}$ as my only solution. However $4e^{\frac{8\pi}{9}i}$ and $4e^{\frac{-4\pi}{9}i}$ are also solutions...

Can anyone see where I have lost my solutions?

2. Mar 14, 2015

### Mentallic

You're missing solutions because you introduced $2k\pi i$ too soon.

$$z^{3/4}=\sqrt{8}e^{\frac{\pi i}{6}}$$

$$z^3=64e^{\frac{2\pi i}{3}}=64e^{\frac{2\pi i}{3}+2k\pi i}$$

and now continue from there.

The reason it doesn't work your way is because you begin with $2k\pi i$ but then you raise each side to the 4th power, hence multiplying $2k\pi i$ by 4, giving you $8k\pi i$ which is obviously missing possible solutions.

3. Mar 14, 2015

### Ray Vickson

To find all the roots of $z^p = W$, take any one root, $r$ and then multiply it by the $p$th roots of unity. Thus, if $\omega \neq 1$ is a root of $\omega^p = 1$, then so are $\omega^{-1}, \omega^{\pm 2}, \omega^{\pm 3}, \ldots$. That gives the other roots of $W$ as $r \omega^{\pm 1}, r \omega^{\pm 2}, \ldots$.