Complex Number Arithmetic...

 

\((1 + i\sqrt{3})^6\\ =\binom{6}{0}(i\sqrt3)^0+\binom{6}{1}(i\sqrt3)^1+\binom{6}{2}(i\sqrt3)^2 +\binom{6}{3}(i\sqrt3)^3+\binom{6}{4}(i\sqrt3)^4+\binom{6}{5}(i\sqrt3)^5+\binom{6}{6}(i\sqrt3)^6\\ =1+6*(i\sqrt3)^1+15(i\sqrt3)^2+20(i\sqrt3)^3+15(i\sqrt3)^4+6(i\sqrt3)^5+(i\sqrt3)^6\\ =1+(6\sqrt3)i-15(\sqrt3)^2-20(\sqrt3)^3i+15(\sqrt3)^4+6(\sqrt3)^5i-(\sqrt3)^6\\ =1+(6\sqrt3)i-15*3-60(\sqrt3)i+15*9+6*9\sqrt3 i-27\\ =1+6\sqrt3i-45-60\sqrt3i+135+54\sqrt3 i-27\\ =1-45+135-27+(6\sqrt3-60\sqrt3+54\sqrt3) i\\ =1-45+135-27+(6\sqrt3-60\sqrt3+54\sqrt3) i\\ =64\)

 

You need to check my working.   there may also have been a simpler path.

 

 

LaTex

(1 + i\sqrt{3})^6\\
=\binom{6}{0}(i\sqrt3)^0+\binom{6}{1}(i\sqrt3)^1+\binom{6}{2}(i\sqrt3)^2
+\binom{6}{3}(i\sqrt3)^3+\binom{6}{4}(i\sqrt3)^4+\binom{6}{5}(i\sqrt3)^5+\binom{6}{6}(i\sqrt3)^6\\
=1+6*(i\sqrt3)^1+15(i\sqrt3)^2+20(i\sqrt3)^3+15(i\sqrt3)^4+6(i\sqrt3)^5+(i\sqrt3)^6\\
=1+(6\sqrt3)i-15(\sqrt3)^2-20(\sqrt3)^3i+15(\sqrt3)^4+6(\sqrt3)^5i-(\sqrt3)^6\\
=1+(6\sqrt3)i-15*3-60(\sqrt3)i+15*9+6*9\sqrt3 i-27\\
=1+6\sqrt3i-45-60\sqrt3i+135+54\sqrt3 i-27\\
=1-45+135-27+(6\sqrt3-60\sqrt3+54\sqrt3) i\\
=1-45+135-27+(6\sqrt3-60\sqrt3+54\sqrt3) i\\
=64

Put the number into polar form, 

\(\displaystyle r(\cos\theta + i\sin\theta) .\)

\(\displaystyle r^{2}=1^{2}+(\sqrt{3})^{2}=4, \text{ }r=2. \)

\(\displaystyle \tan\theta = \sqrt{3}/1, \text{ }\theta= \pi/3.\)

\(\displaystyle (1 + i\sqrt{3})^{6}\\=[2(\cos(\pi/3)+i\sin(\pi/3))]^{6} \\= 2^{6}(\cos(2\pi)+i\sin(2\pi))\\=64.\)