De Moivre's Theorem states that if a complex number is written in the polar coordinate form [ r (cosθ + [tex]i[/tex]sinθ)] and you raise it to the power n, then this can be evaluated by raising the modulus (r) to the power and multiply the argument (θ) by the power. This therefore would give r ⁿ [cos (nθ) + [tex]i[/tex] sin (nθ)].
let A = ∛ (8 cos (4π / 5) + 8 i sin (4π / 5))
⇒ A = ∛ (8 [cos (4π / 5) + i sin (4π / 5)])
Now by applying De Moivre's Theorem,
⇒ A = [tex]8^{ \frac{1}{3} }[/tex] [cos ([tex] \frac{4 \pi }{5} [/tex] × [tex] \frac{1}{3} [/tex]) + [tex]i[/tex] sin ([tex] \frac{4 \pi }{5} [/tex] × [tex] \frac{1}{3} [/tex])
⇒ A = 2 [ cos ([tex] \frac{4 \pi }{15} [/tex]) + [tex]i[/tex] sin ([tex] \frac{4 \pi }{15} [/tex])
⇒ A = 2 [0.0117 + [tex]i[/tex] 0.01297 ] rads
