Question

Find the fourth roots of i

Answer 1

1∠22.5°, 1∠112.5°, 1∠202.5°, 1∠292.5°

Explanation:

A root of a complex number can be found using Euler's identity.

Application

For some z = a·e^(ix), the n-th root is ...

z = (a^(1/n))·e^(i(x/n))

Here, we have z = i, so a = 1 and z = π/2 +2kπ.

Using r∠θ notation, this is ...

i = 1∠(90° +k·360°)

and

i^(1/4) = (1^(1/4))∠((90° +k·360°)/4)

i^(1/4) = 1∠(22.5° +k·90°)

For k = 0 to 3, we have ...

for k = 0, first root = 1∠22.5°

for k = 1, second root = 1∠112.5°

for k = 2, third root = 1∠202.5°

for k = 3, fourth root = 1∠292.5°