what are the cube roots of 8(cos pi/3 + i sin pi/3)

Question

I dont understand how to solve the problem

Answer 1

Use De Moivre's Theorem

 

z^1/3  = 8^1/3(cos π/3 + i sin π/3)^1/3        Three cube roots exist.

 

For the first

 

z^1/3 = 2(cos π/3 + i sin π/3)^1/3

z^1/3 = 2(cos π/9 + i sin π/9 )

z^1/3 = 2(0.110 + 0.006 i)

z^1/3 = 0.055 + 0.003 i

 

For the second

 

z^1/3 = 2(cos 7π/3 + i sin 7π/3)^1/3

z^1/3 = 2(cos 7π/9 + i sin 7π/9)               Complete the calculations

 

For the third

 

z^1/3 = 2(cos 13π/3 + i sin 13π/3)^1/3

z^1/3 = 2(cos 13π/9 + i sin 13π/9)            Complete the calculation