de moivre's theorem

Question

Use De Moivre’s theorem to simplify the expression below. Write your answer in rectangular form a + bi.

[8(cos 7pi/6 + i sin 7pi/6)]^3

Answer 1

make the exponent a coefficient for the angles

[8(cos7pi/6 + isin7pi/6)]^3

=8^3[cos3(7pi/6) + isin3(7pi/6)]

= 512(cos(7pi/2 + isin7pi/2)

= 512(cos3pi/2 + isin3pi/2)

= 512(cos270 + isin270)

=512(0-i)

=-512i

in more explicit a+bi form, it's 0 + (-512)i with a=0 and b=-512

Answer 2

8³ (cos 3π/2 + i sin 3π/2)

Convert to a + bi form