Question

If $\omega$ is an imaginary cube root of unity, then ${{\text{(1+}\omega -{{\omega }^{2}}\text{)}}^{7}}$ equals-

Answer 1

Hint: We have the expression, ${{\text{(1+}\omega -{{\omega }^{2}}\text{)}}^{7}}$ . We need to find the value of this expression. First make the expression in terms of ${{\omega }^{2}}$ . We know the property, $\text{1+}\omega +{{\omega }^{2}}=0$. Using this property, put the value of $\text{1+}\omega =-{{\omega }^{2}}$ and solve it further.

Complete step by step answer:
We know that,
$\text{1+}\omega +{{\omega }^{2}}=0$
$\Rightarrow \text{1+}\omega =-{{\omega }^{2}}$ ……..eq(i)
Putting eq(i) in ${{\text{(1+}\omega -{{\omega }^{2}}\text{)}}^{7}}$
we get,
${{\text{(1+}\omega -{{\omega }^{2}}\text{)}}^{7}}\text{=(}-{{\omega }^{2}}-{{\omega }^{2}}{{\text{)}}^{7}}$
$={{(-2{{\omega }^{2}})}^{7}}=-128{{\omega }^{2.7}}=-128{{\omega }^{14}}$ ………eq(ii)
We know that, ${{\omega }^{3n+2}}={{\omega }^{2}}$
and ${{\omega }^{3n}}=1$
Thus we get,
${{\omega }^{14}}={{\omega }^{3.4}}\times {{\omega }^{2}}=1\times {{\omega }^{2}}={{\omega }^{2}}$ …….eq(iii)
Putting eq.(iii) in eq.(ii),we get
$\text{-128}\times {{\omega }^{14}}=-128\times {{\omega }^{2}}=-128{{\omega }^{2}}$

Hence, ${{\text{(1+}\omega -{{\omega }^{2}}\text{)}}^{7}}=-128{{\omega }^{2}}$ .

Note: In this type of question, one can think to expand the given expression directly after putting the values of $\omega$ and ${{\omega }^{2}}$ . And then it will become complex to solve further and a lot of time can get wasted. To overcome this situation, first, try to make the expression in a single cube root of unity using its property and then expand. After expansion, using the properties of cube roots we can easily solve this question and conclude the answer.