To find the three cube roots of -2 - 2i
find the radius: sqrt( (-2)2 + (-2)2 ) = sqrt( 4 + 4 ) = sqrt(8)
and the angle: tan-1( -2 / -2 ) = 5pi/4 [remember that it is in the third quadrant]
We now have: 81/2cis( 5pi / 4 )
To find the first cube root, we need to find the cube root of 81/2 which is (81/2)1/3 = 81/6
and to divide the angle by 3, giving us 5pi /12.
So, for our first answer, we have 81/6 cis ( 5pi / 12 )
To find the next answer, we need to add 1/3 (or 2pi / 3 ) of the circle to our previous answer: 81/6 cis ( 13pi / 12 )
and to get our last answer we need to add another 2pi / 3 to this answer: 81/6 cis ( 21pi / 12 ) = 81/6 cis ( 7pi / 4 )