Question

The value of ${{\left( 1+\omega -{{\omega }^{2}} \right)}^{7}}$ is
A. $128{{\omega }^{2}}$
B. $-128{{\omega }^{2}}$
C. $128\omega $
D. $-128\omega $

Answer 1

Hint: This type of question can be solved by using the basics of cube roots of unity. This comes under the topic of complex numbers. For this problem, we require the following two important properties,
$1+\omega +{{\omega }^{2}}=0$
${{\omega }^{3}}=1$
Using these equations, we simplify the above expression to the simplest form and evaluate.

Complete step by step solution:
In order to solve this question, we need to know the basics of cube roots of unity. Cube roots of unity means cube roots of 1. These are given as $\omega ,{{\omega }^{2}},{{\omega }^{3}}.$ The value of the cube root ${{\omega }^{3}}$ is 1. The two most important properties required to solve this sum are:
The sum of the cube roots of unity is zero.
$\Rightarrow 1+\omega +{{\omega }^{2}}=0\ldots \ldots \left( 1 \right)$
The cube of an imaginary cube root of unity is 1.
$\Rightarrow {{\omega }^{3}}=1\ldots \ldots \left( 2 \right)$
Now, we need to solve this given equation ${{\left( 1+\omega -{{\omega }^{2}} \right)}^{7}}$ which is in terms of the cube roots of unity. Using the first property, we can rewrite equation (1) by rearranging the terms as,
$\Rightarrow 1+\omega =-{{\omega }^{2}}$
We substitute this value for the value of $1+\omega $ in the given question.
$\Rightarrow {{\left( -{{\omega }^{2}}-{{\omega }^{2}} \right)}^{7}}$
Adding the two terms inside the brackets,
$\Rightarrow {{\left( -2{{\omega }^{2}} \right)}^{7}}$
We now need to calculate the value of the above expression. We know that the value of ${{\left( -2 \right)}^{7}}$ is obtained by multiplying -2 seven times. This value is found out to be -128.
$\Rightarrow -128{{\left( {{\omega }^{2}} \right)}^{7}}$
We know that ${{\left( {{\omega }^{2}} \right)}^{7}}$ is represented as ${{\omega }^{2\times 7}}.$ Therefore using this in the above equation,
$\Rightarrow -128{{\omega }^{14}}$
We now take out one ${{\omega }^{2}}$ term outside from the above expression.
$\Rightarrow -128{{\omega }^{12}}.{{\omega }^{2}}$
The ${{\omega }^{12}}$ term can be written as,
$\Rightarrow -128{{\left( {{\omega }^{3}} \right)}^{4}}.{{\omega }^{2}}$
Using the second property now, we can see that ${{\omega }^{3}}=1.$ Substituting this in the above equation,
$\Rightarrow -128{{\left( 1 \right)}^{4}}.{{\omega }^{2}}$
We know 1 raised to any power will yield 1 itself. Therefore,
$\Rightarrow -128{{\omega }^{2}}$
Therefore, the value of ${{\left( 1+\omega -{{\omega }^{2}} \right)}^{7}}$ is $-128{{\omega }^{2}}.$ Hence, the correct option is B.

So, the correct answer is “Option B”.

Note: To solve this question, the students are required to have a good understanding in the topic of cube roots of unity and their properties. A large number of problems can be solved using these properties. It is suggested to go through all 6 properties of cube roots of unity in order to solve these questions with ease.