Question

Determinati mulțimea A ={n €N|radical din 10 minus radical din n €N}

Answer 1

$\sqrt{10} - \sqrt{n} \in N \Rightarrow \sqrt{10}- \sqrt{n} \geq 0 \Rightarrow \sqrt{10} \geq \sqrt{n} \Rightarrown \leq 10. \\ \\ \sqrt{n} \geq 0 \Rightarrow \sqrt{10} - \sqrt{n} \leq \sqrt{10}\ \textless \ 4 \Rightarrow (\sqrt{10}- \sqrt{n}) \in \{0;1;2;3 \}. \\ \\ \sqrt{10}- \sqrt{n}=0 \Rightarrow \sqrt{10} = \sqrt{n} \Rightarrow n=10. \\ \\ \sqrt{10}- \sqrt{n}=1 \Rightarrow \sqrt{n}= \sqrt{10}-1 \Rightarrow n=(\sqrt{10}-1)^2 \notin N. \\ \\ \sqrt{10}- \sqrt{n}=2 \Rightarrow n=(\sqrt{10 } - 2)^2 \notin N.$

$\sqrt{10}- \sqrt{n} =3 \Rightarrow n= (\sqrt{10} -3)^2 \notin N. \\ \\ Deci~A=\{10\}.$