How do you use demoivre's theorem to simplify #4(1-sqrt3i)^3#?
1 Answer
Explanation:
DeMoivre's theorem for exponents says that any complex number
It continues to say that when raising an imaginary number to a certain power, the result is:
#z^n = r^n color(white)".""cis"(ntheta)#
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To simplify this, let's first calculate
We find
#r# by using Pythagoras' theorem:
#r = sqrt(1^2+sqrt(3)^2) = sqrt4 = 2# We find
#theta# by taking the inverse tangent of#(Im(z))/(Re(z)# .
#theta = tan^-1((-sqrt3)/(1))=-pi/3#
So
#(1-isqrt3)^3 = 2^3"cis"(3(-pi/3)) = 8"cis"(pi)#
Finally, we expand
#8"cis"(pi)=8cospi+8isinpi = 8(-1)+8i(0) = -8#
And don't forget to multiply by 4!
#-8*4 = -32#
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So
Final Answer