Question

Find the value of $ {\left( {1 - \omega + {\omega ^2}} \right)^5} + {\left( {1 - {\omega ^2} + \omega } \right)^5} $ , if $ \omega $ and $ {\omega ^2} $ are complex cube roots of unity.

Answer 1

Hint: First we have to define the terms we need to solve the problem.
Given that $ \omega $ and $ {\omega ^2} $ are complex cube roots of unity.
Therefore,
 $ {\omega ^3} = 1 $
Use the identity,
 $ 1 + \omega + {\omega ^2} = 0 $
Therefore,
 $ 1 + \omega = - {\omega ^2} $
 $ 1 + {\omega ^2} = - \omega $
Formula Used:
 $ {\omega ^3} = 1 $
 $ 1 + \omega + {\omega ^2} = 0 $

Complete step by step answer:
We need to find the value of $ {\left( {1 - \omega + {\omega ^2}} \right)^5} + {\left( {1 - {\omega ^2} + \omega } \right)^5} $
To find the required value let us solve the above equation,
 $ {\left( {1 - \omega + {\omega ^2}} \right)^5} + {\left( {1 - {\omega ^2} + \omega } \right)^5} $
Let us rearrange the terms in the bracket,
 $ = {\left( {1 + {\omega ^2} - \omega } \right)^5} + {\left( {1 + \omega - {\omega ^2}} \right)^5} $
On substituting value of $ 1 + {\omega ^2} = - \omega $ in first bracket we get,
 $ = {\left( { - \omega - \omega } \right)^5} + {\left( {1 + \omega - {\omega ^2}} \right)^5} $
On substituting value of $ 1 + \omega = - {\omega ^2} $ in second bracket we get,
 $ = {\left( { - \omega - \omega } \right)^5} + {\left( { - {\omega ^2} - {\omega ^2}} \right)^5} $
On performing the addition in first bracket we get,
 $ = {\left( { - 2\omega } \right)^5} + {\left( { - {\omega ^2} - {\omega ^2}} \right)^5} $
On performing the addition in second bracket we get,
 $ = {\left( { - 2\omega } \right)^5} + {\left( { - 2{\omega ^2}} \right)^5} $
On performing the power operations of the first bracket we get,
 $ = - 32{\omega ^5} + {\left( { - 2{\omega ^2}} \right)^5} $
On performing the power operations of the second bracket we get,
 $ = - 32{\omega ^5} - 32{\omega ^{10}} $
On taking \[ - 32{\omega ^5}\] from the both brackets we get,
\[ = - 32{\omega ^5}(1 + {\omega ^2})\]
On substituting value of $ 1 + {\omega ^2} = - \omega $ in the bracket we get,
\[ = - 32{\omega ^5} \times ( - \omega )\]
On performing the multiplication of both terms we get,
\[ = 32{\omega ^6}\]
On rearranging the terms we get,
\[ = 32 \times {({\omega ^3})^2}\]
On substituting value of $ {\omega ^3} = 1 $ in the bracket we get,
\[ = 32 \times {(1)^2}\]
On performing the multiplication of both terms we get,
\[ = 32\]
This is the required value.
Therefore,
 $ {\left( {1 - \omega + {\omega ^2}} \right)^5} + {\left( {1 - {\omega ^2} + \omega } \right)^5} = 32 $

Note: Any number when raised to power $ 3 $ or multiplied itself thrice, if given an answer as $ 1 $ , then such number is called the cube root of unity. Complex cube root is usually denoted by $ \omega $ . Complex cube roots of unity are imaginary cube roots of unity. One imaginary cube root of unity is the square of another imaginary cube root of unity i.e. one imaginary cube root of unity is $ \omega $ and other is $ {\omega ^2} $ .