use Demoivre's Theorem to find the indicated power of the complex number. Express the result in standard form.

Question

use Demoivre's Theorem to find the indicated power of the complex number. Express the result in standard form.

(2+2i)^6

a=2 b=2 n=6

r=sqrt 2^2 + 2^2 = sqrt8
Q=7pi/4

(sqrt8)^6 = 512
512(cos 6/1 x 7pi/4) + i sin 6 x 7pi/4
512 (cos 21pi/2) + i sin (21pi/2)
512(cos pi/2) + i (sin pi/2)
512(2+2i)

1024 + 1024i

Is this correct?

Answer 1

Yes, your calculation is correct. Using DeMoivre's Theorem, you correctly found the magnitude (r) of the complex number (2+2i) as √8, and the angle (Q) as 7π/4. Then you raised the magnitude to the power of 6 and multiplied it with the cosine and sine of the product of the angle and the power. Finally, you expressed the result in standard form as 1024 + 1024i. Well done!