Question

Find the three cube roots of the complex number 8i. Give your answers in the form x + iy

Answer 1

if I recall correctly, its the three spokes of a wheel; at 90degrees, 210degrees, and 330 degrees....

Answer 2

how do I change it into x + iy form though?

Answer 3

change it into polar corrdinate form first and use Demovers thrm onit...

Answer 4

8i is:

z = 8(cos(90) + i sin(90)) ; now take the 1/3 power of it...

Answer 5

when we take this to a normal integer power of 3 we get:

z^3 = 8^3(cos(90*3) + i sin(90*3)) right? its the same for rational exponents as well.

Answer 6

z^(1/3) - 2(cos(90/3) + i sin(90/3))

Answer 7

90/3 = 30; the cos(30)=sqrt(3)/2 and the sin(30) = 1/2

2(sqrt(3)/2 + i 1/2) = sqrt(3) + 1i should be one of the cube roots

Answer 8

if I did it right ;)

Answer 9

the others should be at 120 degree intervals from that one... at 150 and 270

Answer 10

(sqrt(3) + i)^2 = 2 +2sqrt(3) i

(sqrt(3)+i) (2 +2sqrt(3) i)
2sqrt(3) + 6i +2i -2sqrt(3) = 8i

Answer 11

that was cool ..... I wonder if im right on the others...

2(cos(270) + i sin(270))

2(0 + i (-1)) = -2i

(-2i)^2 = -4
-4(2i) = -8i ..... Doh!!; well I got the first one right ;)

Answer 12

2(cos(150) + i sin(150))

2(-sqrt(3)/2 + i 1/2)

-sqrt(3) + i; should be another one,

(-sqrt(3) +i)^2 = 3 -1 -2i sqrt(3)

(2 -2i sqrt(3))(-sqrt(3) + i)
-2sqrt(3) +2i +6i +2sqrt(3)

8i ..... well that ones good ;)

Answer 13

0 - (2i) is the last one;

- (2i)^2 = -(4i^2) = -(-4)(2i) = -(-8i) = 8i