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Simplify (1 + i*sqrt3)^6.

 Aug 23, 2023
 #1
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In exponential notation, 1 + i*sqrt(3) = sqrt(3)*e^(i*pi/3).  Then

(1 + i*sqrt(3))^6 = (sqrt(3)*e^(i*pi/3))^6 = 27.

 Aug 23, 2023
 #2
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\(\displaystyle 1 + i\sqrt{3} = 2(1/2 + i\sqrt{3}/2)=2(\cos(\pi/3)+i\sin(\pi/3)). \\ (1+i\sqrt{3})^{6}=\{2(\cos(\pi/3)+i\sin(\pi/3))\}^{6}=2^{6}(\cos(2\pi)+i\sin(2\pi)) \\ =2^{6}=64.\)

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 Aug 24, 2023

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