Step by step
The question
Question: work out (2 + 2i) to the power 4.
Multiplying it out would take forever.
1 Plot it
The complex plane: the real part goes across, the i part goes up.
2 + 2i is the point (2, 2): 2 right, 2 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 = 2 and b = 2.
2 squared is 4.
2 squared is 4.
So r = 2√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 2 + 2i = 2√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√2 to the power 4, and the angle 4 times 45°.
6 Work it out
Careful!
r to the power 4 means 4 of them multiplied: not times 4.
2√2 times 2√2 is 8.
4 of them is 2 such pairs.
So r to the 4 is 64.
4 times 45° is 180°.
So we get 64(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 64: −64.
8 Check
Let's check the size of our answer, by Pythagoras: 64.
It must be r to the power 4: 64.
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.