I'm doing a review for my precalculus final and I don't remember this part

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Directions: Use DeMoivre's theorem to find the indicated power of the complex #. Write the result in standard form. There are two problems:

(3-3i)^6

Find the fourth roots of 128(1+sqrt(3)i)

Jun 4th, 2015

Thank you for the opportunity to help you with your question!

(3-3i)^6 = {3sqrt(2)}^6 (cos (-pi/4)+isin(-pi/4))^6 = 3^6*2^3 *(cos(-3pi/2)+isin(-3pi/2)) = 5832*(0+i) = 5832i

128(1+sqrt(3)i ) = 256 (cospi/3 +isinpi/3)

4th roots are 

256^(1/4) (cos pi/12+i sin pi/12) =2 (cos pi/12+i sin pi/12) = {sqrt(6)+sqrt(2) +i(sqrt(6)-sqrt(2))}/2

2 (cos (pi/12+pi/2)+i sin (pi/12+pi/2)) =  {-sqrt(6)-sqrt(2) +i(sqrt(6)+sqrt(2))}/2

2 (cos (pi/12+pi)+i sin (pi/12+pi)) = -{sqrt(6)+sqrt(2) +i(sqrt(6)-sqrt(2))}/2

2 (cos (pi/12+3pi/2)+i sin (pi/12+3pi/2)) = {sqrt(6)+sqrt(2) -i(sqrt(6)+sqrt(2))}/2




Please let me know if you need any clarification. I'm always happy to answer your questions.
Jun 4th, 2015

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