Question

Para que valores de n se cumpre a igualdade (1+i√3)ⁿ=(1-i√3)ⁿ ?​

Answer 1

$\boxed{\begin{array}{l}\sf(1+i\sqrt{3})^n=(1-i\sqrt{3})^n\\\sf\ell n(1+i\sqrt{3})^n=\ell n(1-i\sqrt{3})^n\\\sf lembre-se~que~\ell og_ba^c=c\cdot\ell o_ba\\\sf n\ell n(1+i\sqrt{3})=n\ell n(1-i\sqrt{3})\\\sf n\ell n(1+i\sqrt{3})-n\ell n(1-i\sqrt{3})=0\\\rm colocando~n~em~evid\hat encia~temos:\\\sf n[\ell n(1+i\sqrt{3})-\ell n(1-i\sqrt{3})]=0\\\underline{\rm Propriedade~do~produto~nulo}\\\boldsymbol{ a\cdot b=0\Longleftrightarrow a=0~ou~b=0}\\\sf como~os~logaritmos~n\tilde ao~se~anulam\\\sf resta~dizer~que~ n=0\\\sf portanto\\\sf (1+i\sqrt{3})^n=(1-i\sqrt{3})^n\Longleftrightarrow n=0\end{array}}$