Question

If $\omega $ is a cube root of unity, then the value of ${\left( {1 - \omega + {\omega ^2}} \right)^5} + {\left( {1 + \omega - {\omega ^2}} \right)^5}$.
A. $30$
B. $32$
C. $2$
D. None of these

Answer 1

Hint: In the given question, we are given an expression involving $\omega $ as cube root of unity. So, we have to find the value of the given expression using the properties of cube roots of unity. We will use simplification rules to simplify the value of the expression and get to the required answer. We must know the value of the expression $\left( {1 + \omega + {\omega ^2}} \right)$ in order to solve the question.

Complete step by step answer:
The given expression involving the cube root of unity is ${\left( {1 - \omega + {\omega ^2}} \right)^5} + {\left( {1 + \omega - {\omega ^2}} \right)^5}$. Now, we know that the value of the expression $\left( {1 + \omega + {\omega ^2}} \right)$ is zero. This is the basic property of the cube root of unity. So, we will use this property to simplify the value of the expression.
$ \Rightarrow {\left( {1 - \omega + {\omega ^2}} \right)^5} + {\left( {1 + \omega - {\omega ^2}} \right)^5}$
Now, we get the value of $\left( {{\omega ^2} + 1} \right)$ as $\left( { - \omega } \right)$ from $\left( {1 + \omega + {\omega ^2}} \right) = 0$. Also, $\left( {1 + {\omega ^2}} \right) = - \omega $. So, we get,
\[ \Rightarrow {\left( { - \omega - \omega } \right)^5} + {\left( { - {\omega ^2} - {\omega ^2}} \right)^5}\]

Adding up the like terms, we get,
\[ \Rightarrow {\left( { - 2\omega } \right)^5} + {\left( { - 2{\omega ^2}} \right)^5}\]
Calculating the powers, we get,
\[ \Rightarrow \left( { - 32{\omega ^5}} \right) + \left( { - 32{\omega ^{10}}} \right)\]
Opening the brackets and taking \[32\] common from both the terms, we get,
\[ \Rightarrow - 32\left( {{\omega ^5} + {\omega ^{10}}} \right)\]
Now, we know that $\omega $ and ${\omega ^2}$ are the cube roots of unity. So, the value of ${\omega ^3}$ is $1$.
\[ \Rightarrow - 32\left( {\left( {{\omega ^3}} \right) \times {\omega ^2} + {{\left( {{\omega ^3}} \right)}^3} \times \omega } \right)\]
\[ \Rightarrow - 32\left( {\left( 1 \right) \times {\omega ^2} + {{\left( 1 \right)}^3} \times \omega } \right)\]
\[ \Rightarrow - 32\left( {{\omega ^2} + \omega } \right)\]
Now, we know that $\left( {1 + \omega + {\omega ^2}} \right) = 0$. So, we get the value of \[\left( {{\omega ^2} + \omega } \right)\] as $\left( { - 1} \right)$.
\[ \Rightarrow - 32\left( { - 1} \right)\]
\[ \Rightarrow 32\]
So, the value of the expression ${\left( {1 - \omega + {\omega ^2}} \right)^5} + {\left( {1 + \omega - {\omega ^2}} \right)^5}$ is $32$.

Therefore, option B is the correct answer.

Note: We must have a good grip over the concepts of complex numbers and the topic of cube root of unity to solve the given problem. One should take care of the calculations while doing such problems so as to be sure of the final answer. We should remember the value of the expression as it can be used as a direct result in various questions.