Question

Vă rog, ajutați-mă cu 1.

Answer 1

$Pentru~ca~radicalii~sa~fie~de.finiti, ~trebuie~ca~a,b\geq 0.\\ \\ Daca~a=b=0~\Rightarrow~evident!\\ \\ Daca~cel~mult~un~numar~este~egal~cu~0,~rezulta~ca~\sqrt{a}+\sqrt{b}\neq 0\\ \\ a~si~b~sunt~rationale~\Rightarrow~a-b\in~Q\\ \\ a-b=(\sqrt{a}-\sqrt{b})(\sqrt{a}+\sqrt{b})~\Rightarrow~\sqrt{a}-\sqrt{b}=\frac{a-b}{\sqrt{a}+\sqrt{b}} \\ \\ Daca~a-b~si~\sqrt{a}+\sqrt{b}~sunt~rationale~\Rightarrow~\frac{a-b}{\sqrt{a}+\sqrt{b}} \in~Q~\Leftrightarrow~\sqrt{a}-\sqrt{b}\in~Q\\ \\ Cum~\sqrt{a}+\sqrt{b}~si~\sqrt{a}-\sqrt{b}~apartin~lui~Q,~rezulta~ca~\\ \\ (\sqrt{a}+\sqrt{b})~+~(\sqrt{a}-\sqrt{b})~=2\sqrt{a}\in~Q.\Leftrightarrow~\sqrt{a}\in~Q.\\ \\ Cum~\sqrt{a}\in~Q~si~\sqrt{a}+\sqrt{b}~\in~Q~\Rightarrow~\sqrt{b}\in~Q.$