How do you find the power #(sqrt3-i)^3# and express the result in rectangular form?
1 Answer
Mar 7, 2017
Explanation:
Let
First let us plot the point
And we will put the complex number into polar form:
# |z| = sqrt(3+1) = 2 #
# arg(z) = tan^-1(-1/sqrt(3)) = -(pi)/6 #
So then in polar form we have:
# z = 2(cos(-(pi)/6) + isin(-(pi)/6)) #
By De Moivre's Theorem:
# z^3 = {2(cos(-(pi)/6) + isin(-(pi)/6))}^3 #
# \ \ \ = 2^3(cos(-(3pi)/6) + isin(-(3pi)/6)) #
# \ \ \ = 8(cos(-(pi)/2) + isin(-(pi)/2)) #
# \ \ \ = 8(0-i) #
# \ \ \ = -8i #