Question

[(1+i)(1+1/i)]^7 Find the solution of given question

Answer 1

$\large\underline{\sf{Solution-}}$

Given

$\rm :\longmapsto\:\bigg[(1 + i)\bigg(1 + \dfrac{1}{i}\bigg)\bigg]^{7}$

We know,

$\red{\rm :\longmapsto\:i = \sqrt{ - 1} }$

$\red{\rm :\longmapsto\: {i}^{2} = - 1}$

$\red{\rm :\longmapsto\:i.i = - 1}$

$\red{\rm :\longmapsto\:\dfrac{1}{i} = - i}$

Now,

The given expression can be rewritten as

$\rm \:  =  \:  \: {\bigg[(1 + i)(1 - i)\bigg]}^{7}$

We know

$\boxed{ \bf{ \: (x + y)(x - y) = {x}^{2} - {y}^{2}}}$

So, using this identity, we get

$\rm \:  =  \:  \: \bigg[1 - {i}^{2}\bigg]^{7}$

We know,

$\boxed{ \bf{ \: {i}^{2} = - \: 1}}$

$\rm \:  =  \:  \: \bigg[1 - ( - 1)\bigg]^{7}$

$\rm \:  =  \:  \: \bigg[1 + 1\bigg]^{7}$

$\rm \:  =  \:  \: \bigg[2\bigg]^{7}$

$\rm \:  =  \:  \: 128$

Hence,

$\: \: \: \: \: \: \: \: \underbrace{\boxed{ \bf{ \: \:\bigg[(1 + i)\bigg(1 + \dfrac{1}{i}\bigg)\bigg]^{7} = 128}}}$

Additional Information :-

Cube roots of unity

$\red{\rm :\longmapsto\:1, \omega, {\omega}^{2} \: are \: cube \: roots \: of \: unity \: such \: that}$

$\boxed{ \bf{ \: 1 + \omega + {\omega}^{2} = 0}}$

$\boxed{ \bf{ \: 1 + \omega = - \: {\omega}^{2} }}$

$\boxed{ \bf{ \: 1 + {\omega}^{2} = - \: \omega}}$

$\boxed{ \bf{ \: \omega + {\omega}^{2} = - \: 1}}$

$\boxed{ \bf{ \: {\omega}^{3} = 1}}$

$\boxed{ \bf{ \: \omega \: = \: \dfrac{ - 1 \: + \: \sqrt{3} \: i}{2}}}$

$\boxed{ \bf{ \: \omega {}^{2} \: = \: \dfrac{ - 1 \: - \: \sqrt{3} \: i}{2}}}$