# Thread: find the cube roots

1. ## find the cube roots

I am learning how to find cube roots of a complex number and I wanted to see if I am doing this right. could someone let me know?

find the cube roots of -8i:

$
8(cos (3\pi/2)+ i sin(3\pi/2))
$

$
(8)^1/3[cos((3\pi/2)+2\pi(k)/3) + i sin((3\pi/2)+2\pi(k)/3)]
$

$
k= 0, 1, 2
$

$
2(cos(\pi/2) + i sin(\pi/2))
$

Is this good so far? I just done the first one to see if I done it right.

2. Dear robasc,

It's not trully correct. Check it cubing for k=1 or 2.

3. Hello, robasc!

I'm pretty sure you've got it right . . . I'll tidy-up a bit just to make sure.

Find the cube roots of $-8i$

$z\;=\;8\left(\cos\frac{3\pi}{2}+ i\sin\frac{3\pi}{2}\right)$

$z^{\frac{1}{3}} \;=\;8^{\frac{1}{3}}\bigg[\cos\left(\frac{\frac{3\pi}{2}+2\pi k}{3}\right) + i\sin\left(\frac{\frac{3\pi}{2}+2\pi k}{3}\right)\bigg] \;=$ . $2\bigg[\cos\left(\frac{\pi}{2} + \frac{2\pi}{3}k\right) + i\sin\left(\frac{\pi}{2} + \frac{2\pi}{3}k\right) \bigg]$

. . for $k\:=\: 0, 1, 2$

$2\left(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\right)$

Is this good so far? I just done the first one to see if I done it right.

Looks good to me!

4. do you mean check for the rest of them?

$
2(cos(7\pi/6)+ i sin(7\pi/6))
$

$
2(cos(11\pi/6)+ i sin(11\pi/6))
$

5. Okey, than you only misspeled the parenthesis but though right.

6. Thanks you friends!