Solve \[4 + 5{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{334}} + 3{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{365}}\] where, $i = \sqrt { - 1} $.
(a) $1$
(b) $i\sqrt 2 $
(c) $i$
(d) \[\sqrt 3 {\text{ }}i\]

Hint: We will be going to use the most curious concept of complex relations to be recognised by those three different formulae of “cube root of unity” containing omega and its respective value, for the analysis of desired expression.

Complete step by step answer:
$\because $ The condition is related to the complex number as there exists the parameter ‘i’ where the value of instance ‘i’ is $i = \sqrt { - 1} $ respectively.
Here, we have given the expression as \[4 + 5{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{334}} + 3{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{365}}\]
So, pre-assuming the term$\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right) = \omega $, this is the complex term.
The given equation becomes,
$4 + 5{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{334}} + 3{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{365}} = 4 + 5{\omega ^{334}} + 3{\omega ^{335}}$
Therefore, using the complex algebraic relation for solution we can reach to the desired value.
Since, using the rules of indices, adjusting the powers so that we get the respective output as per the terminology,
\[4 + 5{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{334}} + 3{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{365}} = 4 + 5{\left( {{\omega ^3}} \right)^{111}}\omega + 3{\left( {{\omega ^3}} \right)^{121}}{\omega ^2}\] \[4 + 5{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{334}} + 3{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{365}} = 4 + 5\omega + 3{\omega ^2}\]
$\because $We know that, ${\omega ^3} = 1$
Now, since expanding the terms to the desired formula, we get
\[4 + 5{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{334}} + 3{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{365}} = 1 + 3 + 2\omega + 3\omega + 3{\omega ^2}\]
Rearranging the terms to the efficient grouping, we get
\[4 + 5{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{334}} + 3{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{365}} = 1 + 2\omega + 3 + 3\omega + 3{\omega ^2}\]
Now, taking the common terms, we get
\[4 + 5{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{334}} + 3{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{365}} = 1 + 2\omega + 3(1 + \omega + {\omega ^2})\]
Substituting the desire value in place of respective formula formed that is, \[1 + \omega + {\omega ^2} = 0\], we get
\[4 + 5{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{334}} + 3{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{365}} = 1 + 2\omega + 0\]
Now, replacing the value of omega in place of the solution that we have assumed previously in the solution, we get
\[4 + 5{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{334}} + 3{\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)^{365}} = 1 + 2\left( { - \dfrac{1}{2} + i\dfrac{{\sqrt 3 }}{2}} \right)\]
Simplifying the above equation after putting the assumed value, we get
\[4 + 5{\left( { - \dfrac{1}{2} + i\sqrt {\dfrac{3}{2}} } \right)^{334}} + 3{\left( { - \dfrac{1}{2} + i\sqrt {\dfrac{3}{2}} } \right)^{365}} = 1 + - 1 + i\sqrt 3 \]
\[4 + 5{\left( { - \dfrac{1}{2} + i\sqrt {\dfrac{3}{2}} } \right)^{334}} + 3{\left( { - \dfrac{1}{2} + i\sqrt {\dfrac{3}{2}} } \right)^{365}} = i\sqrt 3 \]\[ = \sqrt 3 i\]
$\therefore $ The required answer is $ - 1$. The correct option is (b).

Note:
 Considering the three cube roots of unity i.e. $1$,$\omega $ and ${\omega ^2}$ where, \[\omega = \dfrac{{ - 1 + i\sqrt 3 }}{2}\] and \[{\omega ^2} = \dfrac{{ - 1 - i\sqrt 3 }}{2}\]. Since, we also know that, $1 + \omega + {\omega ^2} = 0$ and ${\omega ^3} = 1$. As a result, use the respective formulae and conditions in the calculations to find the desired output (before starting the solutions assume \[\dfrac{{ - 1 + i\sqrt 3 }}{2}\] as $\omega $. Use the proper conditions for the indices and algebraic rules for simplification of expression.