Question

For z=2cis60°, find z^4 in rectangular form

Answer 1

To find z^4, we can raise z to the fourth power, since z^4 = (z)^4. Using the polar form of z, we have:

z = 2cis(60°)

We can convert this to rectangular form using the following formulas:

x = r cosθ

y = r sinθ

where r is the magnitude of z, and θ is the angle in radians.

r = 2

θ = 60° = π/3 radians

x = 2 cos(π/3) = 1

y = 2 sin(π/3) = √3

Therefore, z = 1 + √3i

Now, we can raise z to the fourth power:

z^4 = (1 + √3i)^4

Using the binomial expansion formula, we can expand this expression:

z^4 = 1 + 4(√3i) + 6(-1) + 4(-√3i) + 1

Simplifying, we get:

z^4 = -8 + 8√3i

Therefore, z^4 in rectangular form is -8 + 8√3i.