Step by step
The question
Question: work out (1 − i) to the power 6.
Multiplying it out would take forever.
1 Plot it
The complex plane: the real part goes across, the i part goes up.
1 − i is the point (1, −1): 1 right, 1 down.
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 = −1.
1 squared is 1.
−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 45°, from the unit circle.
The point is in quadrant IV: 360° − 45°.
So θ = 315°.
Careful!
The calculator gives −45°.
That is the same direction as 315°.
We keep θ from 0° to 360°.
4 Write it in polar form
Polar form is r(cos θ + i sin θ).
So 1 − i = √2(cos 315° + i sin 315°).
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 6, and the angle 6 times 315°.
6 Work it out
√2 times √2 is 2.
6 of them is 3 such pairs.
So r to the 6 is 8.
6 times 315° is 1890°.
Full turns change nothing: take away 5 full turns.
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 6: 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.