\section*{Exercise 3 - 031043}

Given the numbers \(Z_{1}=\sqrt{7}+\sqrt{10}\) and \(Z_{2}=\sqrt{3}+\sqrt{19}\).

Determine which of the two numbers is larger without calculating the square roots!
}

It holds that \(27<20 \cdot 7=20 \sqrt{49}<20 \sqrt{57}\). Therefore,

\[
(2 \sqrt{70})^{2}=280=25+27+228<25+20 \sqrt{57}+4 \cdot 57=(5+2 \sqrt{3 \cdot 19})^{2}
\]

Thus, \(2 \sqrt{7 \cdot 10}<5+2 \sqrt{3 \cdot 19}\) or

\[
(\sqrt{7}+\sqrt{10})^{2}=7+2 \sqrt{7 \cdot 10}+10<3+2 \sqrt{3 \cdot 19}+19=(\sqrt{3}+\sqrt{19})^{2}
\]

From this, it follows that \(\sqrt{7}+\sqrt{10}<\sqrt{3}+\sqrt{19}\) and \(Z_{1}<Z_{2}\).