Step by step
The question
Question: work out (−√3 + i) to the power 3.
Multiplying it out would take forever.
1 Plot it
The complex plane: the real part goes across, the i part goes up.
−√3 + i is the point (−1.73, 1): 1.73 left, 1 up.
r is its distance from 0.
θ is its angle, from the positive real axis, turning counterclockwise.
2 Find r
Pythagoras: r = √(a² + b²), with a = −√3 and b = 1.
−√3 squared is 3.
1 squared is 1.
So r = 2.
3 Find θ
The reference angle: tan is up over across, without the signs.
That is the tan of 30°, from the unit circle.
The point is in quadrant II: 180° − 30°.
So θ = 150°.
Careful!
The calculator's tan⁻¹ only gives angles in quadrants I and IV.
Our point is in quadrant II: look at the picture.
4 Write it in polar form
Polar form is r(cos θ + i sin θ).
So −√3 + i = 2(cos 150° + i sin 150°).
5 Use De Moivre
De Moivre's rule: to raise it to the power n, raise r to the power n, and multiply the angle by n.
So r becomes 2 to the power 3, and the angle 3 times 150°.
6 Work it out
2 to the power 3 is 8.
So r to the 3 is 8.
3 times 150° is 450°.
Full turns change nothing: take away 360°.
90° points the same way.
So we get 8(cos 90° + i sin 90°).
7 Back to a + bi
From the unit circle: cos 90° is 0, and sin 90° is 1.
Multiply by 8: 8i.
8 Check
Let's check the size of our answer, by Pythagoras: 8.
It must be r to the power 3: 8.
The same!
So it is right.
Let's look back.
First: plot it.
Then: find r.
Then: find θ.
Then: write it in polar form.
Then: use De Moivre.
Then: work it out.
Then: back to a + bi.
Last: check.