Question
If $$i = \sqrt { - 1} ,{\text{ then }}4 + 5{\left( { - \frac{1}{2} + \frac{{i\sqrt 3 }}{2}} \right)^{334}} + 3{\left( { - \frac{1}{2} + \frac{{i\sqrt 3 }}{2}} \right)^{365}}$$ is equal to
A.
$$1 - i\sqrt 3 $$
B.
$$ - 1 + i\sqrt 3 $$
C.
$$i\sqrt 3 $$
D.
$$ - i\sqrt 3 $$
Answer :
$$i\sqrt 3 $$
Solution :
$$\eqalign{
& E = 4 + 5{\left( \omega \right)^{334}} + 3{\left( \omega \right)^{365}} = 4 + 5\omega + 3{\omega ^2} \cr
& \,\,\,\,\,\, = 1 + 2\omega + 3\left( {1 + \omega + {\omega ^2}} \right) = 1 + \left( { - 1 + i\sqrt 3 } \right) \cr
& \,\,\,\,\,\, = \,\,i\sqrt 3 \cr} $$