Step by step
The question
Question: work out (1 + i) to the power 8.
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 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 = 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 I, so θ = 45°.
4 Write it in polar form
Polar form is r(cos θ + i sin θ).
So 1 + i = √2(cos 45° + i sin 45°).
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 8, and the angle 8 times 45°.
6 Work it out
Careful!
r to the power 8 means 8 of them multiplied: not times 8.
√2 times √2 is 2.
8 of them is 4 such pairs.
So r to the 8 is 16.
8 times 45° is 360°.
Full turns change nothing: take away 360°.
0° points the same way.
So we get 16(cos 0° + i sin 0°).
7 Back to a + bi
From the unit circle: cos 0° is 1, and sin 0° is 0.
Multiply by 16: 16.
8 Check
Let's check the size of our answer, by Pythagoras: 16.
It must be r to the power 8: 16.
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.