Question

Use Demoivre's theorem to find the indicated power of the complex number. Write the result in standard form.

1.(3(cos5pi/4+i sin 5pi/4))^8
2.(3-3i)^6

Answer 1

(3(cos(5pi/4)+i sin (5pi/4) ) )^8
= 3^8(cos 8(5π/4) + i sin 8(5π/4))
= 6561( cos 10π + i sin 10π)
= 6561( 1 + i(0) )
= 6561

B
(3-3i)^6
let z = 3-3i, so we want z^6

r = √(9 + 9) = √18 = 3√2
plot (3,-3) which is in IV, so
Ø = 315° or 7π/4
z = 3√2(cos 7π/4 + i sin 7π/4)
z^6 = (3√2)^6 [cos 42π/4 + i sin 42π/4]
= 5832( 0 + i (1) )
= 5832 i