The least positive integer n for which (1+i√31-i√3)^n=1, is

Question

The least positive integer $n$ for which $\left(\dfrac{1+i\sqrt3}{1-i\sqrt3}\right)^n=1$, is

Accepted Answer

$${\left( {{{1 + i\sqrt 3 } \over {1 - i\sqrt 3 }}} \right)^n} = 1$$

$$ \Rightarrow \,\,\,\,\,{\left( {{{{{1 + i\sqrt 3 } \over 2}} \over {{{1 - i\sqrt 3 } \over 2}}}} \right)^n} = 1$$

We know,  $$\omega $$ = - $${{1 - i\sqrt 3 } \over 2}$$

and $$\omega $$2 = - $${{\left( {1 + i\sqrt 3 } \right)} \over 2}$$

$$ \Rightarrow $$ $$\,\,\,\,$$ $${\left( {{{ - {\omega ^2}} \over { - \omega }}} \right)^n} = 1$$

$$ \Rightarrow $$ ($$\omega $$)n = 1 = $$\omega $$3

$$ \Rightarrow $$ $$\,\,\,\,$$ n = 3

Solutions · 1