Step by step
The question
Question: work out (−1 + i) 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.
−1 + i is the point (−1, 1): 1 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 = −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 II: 180° − 45°.
So θ = 135°.
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 −1 + i = √2(cos 135° + i sin 135°).
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 4, and the angle 4 times 135°.
6 Work it out
√2 times √2 is 2.
4 of them is 2 such pairs.
So r to the 4 is 4.
4 times 135° is 540°.
Full turns change nothing: take away 360°.
180° points the same way.
So we get 4(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 4: −4.
8 Check
Let's check the size of our answer, by Pythagoras: 4.
It must be r to the power 4: 4.
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.