Simplify $(-i)^\frac{1}{4}$

Question

How do I simplify $(-i)^\frac{1}{4}$? I know that it has norm 1 and I think that the angle is $-\frac{\pi}{4}$, but I'm not really sure.

Answer 1

Does this figure help you (a plot in the complex plane), where the red arrow points to $-i$? The identity ($1$) is to the right and each multiplication rotates the identity to the next arrow.

How many steps of rotation are there?

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(Drawn in Mathematica)

Answer 2

Write $-i$ in polar form. $$-i=e^{\frac{3\pi i}{2}}$$ Thus $$(-i)^{\frac{1}{4}}=e^{\frac{3\pi i}{8}+\frac{k\pi}{2}}$$ where $k \in \{ 0,1,2,3 \}$.

Answer 3

Note that

$$-i=e^{3i\pi/2+2k\pi}$$

for all integers $k$.

Now take the 4th root...

$$(-i)^{1/4}=e^{3i\pi/8+k\pi/2}=\cos{\left(\frac{3\pi}{8}+\frac{k\pi}{2}\right)}+i\sin{\left(\frac{3\pi}{8}+\frac{k\pi}{2}\right)}$$

If you want your expression in rectangular coordinates (since that is how you started), you could expand sine and cosine using the addition formulas and the half angle identities as

$$\frac{3\pi}{8}=\frac{1}{2}\frac{3\pi}{4}$$

It will be a little messier but you'd have another form of the solution.