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Aratati ca numarul a= I[tex] \sqrt{11} -3[/tex] I + [tex] \frac{2}{3- \sqrt{11} } [/tex] este intreg

Va rog din suflet!


Răspuns :

a=I√11-3I + 2/(3-√11)=
=√11-3+2(3+√11)/(3-√11)(3+√11)=
=√11-3+2(3+√11)/(3-11)=
=√11-3+2(3+√11)/(-2)=
=√11-3-(3+√11)=
=√11-3-3-√11)=-6
√11=3,...>3, deci I√11-3I=√11-3
√11-3+2(3+√11)/(3-√11)(3+√11)=√11-3+2(3+√11)/(9-11)=
√11-3+2(3+√11)/(-2)=√11-3-3-√11=-6∈Z