Question

$Fie~a,b\in~(0,inf)~si~z,u\in~C. \\ z=( \sqrt{a} +i \sqrt{b} )( \sqrt{a} +i^2 \sqrt{b} )( \sqrt{a} +i^3 \sqrt{b}) *...*( \sqrt{a} +i^{4n} \sqrt{b} ) \\ u=( \sqrt[4]{a} -i \sqrt[4]{b} )( \sqrt[4]{a} - i^2\sqrt[4]{b} )( \sqrt[4]{a} -i^3 \sqrt[4]{b} )*..*( \sqrt[4]{a} -i^{4n} \sqrt[4]{b} ) \\ Sa~se~arate~ca~z,u~sunt~numere~reale.$

Answer 1

Z   se   mai   scrie
z=(√a+ib)(√a-√b)(√a-i√b)(√a+i^4√b)*,,,*(√a+i^4n)=
aranjezi   ptimii   4   factori   convenabil=
(√a+i√b)*(√a-i√b)(√a-√b)*(√a+√b)*...(√a+i^4n)=
(a+b)(a-b)*...*(√a+i^4n√b)=
(a²-b²)*...(√a+i^4n√b)=
fiecare   secventa   de   4   paranteze   este   egala  cu (a²-b²) deci   
z=(a²-b²)^n∈R

u=(√a-i√b)(√a+√b)(√a+i√b)(√a+√b)....*(√a+i^4n√b)=  √a  radical ordinul   4   din  a  √b=radical ordinul   4   din   b