Please help , need to find cube roots of a number?

Question

find all roots of -3+ 3 j , in exact polar form . How many inequivalent roots does this complex number have . Which root has the smallest argument. Thanks a lot for your help .

Never mind , i worked it out

Answer 1

The angle of the vector v = -3 + 3i is 135°. So the reference angle for the cube root is 135/3 = 45°.

The remaining two cube roots are equally spaced out around the circle at

45 + 120 = 165°
45 + 240 = 285°

The magnitude of v is:
|| v || = √[(-3)² + 3²] = √(9 + 9) = √18 = 18^(1/2)

The magnitude of the cube root is:
[18^(1/2)]^(1/3) = 18^(1/6)

The three cube roots are:

(i)
18^(1/6)*(cos45° + i sin45°)
= 18^(1/6) [1/√2 + i/√2]
= [(3/2)^(1/3) + i (3/2)^(1/3)]

(ii)
18^(1/6)*(cos165° + i sin165°)

(iii)
18^(1/6)*(cos285° + i sin285°)
________

In polar form that would be:

(r, θ) = (18^(1/6), 45°); (18^(1/6), 165°); and (18^(1/6), 285°)

Answer 2

The sq. root of a unfavourable form is imaginary, it quite is, it does not exist in our international. by using the regulations of arithmetic that we shop on with, any form that's squared, unfavourable or effective leads to a favorable answer. The sq. root of four is the two +2 or -2. the easy rule is that comparable warning signs extra desirable consequence in a favorable answer. Cubes are the consequence of three multiplications. the respond could have the sign of the backside form. as an occasion, -3 x -3 = +9. +9 x -3 = -27. The sq. root of any form is the two effective or unfavourable. The cube root of any form would be the same sign because of the fact the form.

Answer 3

Man, that would keep me up all night, too!