Step by step
The question
Question: work out (1 + √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.
1 + √3 i is the point (1, 1.73): 1 right, 1.73 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 = 1 and b = √3.
1 squared is 1.
√3 squared is 3.
So r = 2.
3 Find θ
The reference angle: tan is up over across, without the signs.
That is the tan of 60°, from the unit circle.
The point is in quadrant I, so θ = 60°.
4 Write it in polar form
Polar form is r(cos θ + i sin θ).
So 1 + √3 i = 2(cos 60° + i sin 60°).
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 60°.
6 Work it out
Careful!
r to the power 3 means 3 of them multiplied: not times 3.
2 to the power 3 is 8.
So r to the 3 is 8.
3 times 60° is 180°.
So we get 8(cos 180° + i sin 180°).
7 Back to a + bi
From the unit circle: cos 180° is −1, and sin 180° is 0.
Multiply by 8: −8.
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.