Simplify (√3 + i)^6 Step by Step

Step by step

The question

Question: work out (√3 + i) to the power 6.

Multiplying it out would take forever.

(√3 + i)⁶ =

1 Plot it

The complex plane: the real part goes across, the i part goes up.

√3 + i is the point (1.73, 1): 1.73 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 = √3 and b = 1.

r = √(√3)² + 1²

√3 squared is 3.

1 squared is 1.

So r = 2.

r = √3 + 1 = √4 = 2

3 Find θ

The reference angle: tan is up over across, without the signs.

That is the tan of 30°, from the unit circle.

tan (ref) = 1√3 → 30°

The point is in quadrant I, so θ = 30°.

θ = 30°

4 Write it in polar form

Polar form is r(cos θ + i sin θ).

So √3 + i = 2(cos 30° + i sin 30°).

= √3 + i = 2(cos 30° + i sin 30°)

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.

[r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ)

So r becomes 2 to the power 6, and the angle 6 times 30°.

(√3 + i)⁶ = 2⁶(cos(6 × 30°) + i sin(6 × 30°))

6 Work it out

Careful!

r to the power 6 means 6 of them multiplied: not times 6.

Careful: not 2 × 6 = 12 ✗

2 to the power 6 is 64.

So r to the 6 is 64.

(2)⁶ = 64

6 times 30° is 180°.

6 × 30° = 180°

So we get 64(cos 180° + i sin 180°).

= 64(cos 180° + i sin 180°)

7 Back to a + bi

From the unit circle: cos 180° is −1, and sin 180° is 0.

= 64(−1 + 0i)

Multiply by 64: −64.

(√3 + i)⁶ = −64

8 Check

Let's check the size of our answer, by Pythagoras: 64.

size = √4096 + 0 = 64

It must be r to the power 6: 64.

The same!

So it is right.

(2)⁶ = 64

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.

Answer