• question_answer $\omega$ is an imaginary cube root of unity. If ${{(1+{{\omega }^{2}})}^{m}}=$ ${{(1+{{\omega }^{4}})}^{m}},$ then least positive integral value of m is  [IIT Screening 2004] A) 6 B) 5 C) 4 D) 3

We have, ${{(1+{{\omega }^{2}})}^{m}}={{(1+{{\omega }^{4}})}^{m}}$ $(\because \,\,{{\omega }^{3}}=1)$ ${{(1+{{\omega }^{2}})}^{m}}={{(1+\omega )}^{m}}$ ${{(-\omega )}^{m}}={{(-{{\omega }^{2}})}^{m}}$ $\Rightarrow {{\left( \frac{\omega }{{{\omega }^{2}}} \right)}^{m}}=1$$\Rightarrow {{({{\omega }^{2}})}^{m}}=1$$={{(\omega )}^{2m}}=({{\omega }^{3}})$$\Rightarrow m=\frac{3}{2}$ Hence least positive integral value of m is 3.