Hello! Help me please, how to solve this example? (1 + i sqrt 3)^40

1 Answer
Mar 21, 2018

#-2^39(1+isqrt{3})#

Explanation:

It is easy to solve problems of this kind if you use the polar representation of complex numbers

#1+isqrt{3} = 2(1/2+i sqrt{3}/2) = 2(cos (pi/3)+i sin(pi/3))#

Now, we can use
#(cos theta_1 +i sin theta_1)(cos theta_2 +i sin theta_2)=cos (theta_1+theta_2) +i sin (theta_1+theta_2)#
to prove
#(cos theta +i sin theta)^n = cos (n theta) +i sin (n theta)#

Thus

#(1+isqrt{3})^{40} = (2(cos (pi/3)+i sin(pi/3)))^40 = 2^40(cos (pi/3)+i sin(pi/3))^40 = 2^40(cos (40pi/3)+i sin(40pi/3)) = 2^40(cos (12pi+ (4pi)/3)+i sin(12pi +(4pi)/3)) = 2^40(cos ((4pi)/3)+i sin((4pi)/3)) = 2^40(-1/2-isqrt{3}/2) = -2^39(1+isqrt{3})#