[tex]\displaystyle \bf \\
\sqrt{49 + 20 \sqrt{6}} + \sqrt{49 - 20 \sqrt{6}}=\\\\
= \sqrt{49 + 2 \times 10 \sqrt{6}} + \sqrt{49 - 2\times 10 \sqrt{6}}=\\\\
= \sqrt{49 + 2 \times 5\times 2\sqrt{6}} + \sqrt{49 - 2\times 5 \times 2\sqrt{6}}\\\\
\text{Observam ca: }~~ 5^2 + (2\sqrt{6})^2 = 25 + 24 = 49
[/tex]
[tex]\displaystyle\bf\\
\Longrightarrow~\text{Vom descompune pe 49 astfel: }49=25+24\\\\
\sqrt{49+2\times5\times2\sqrt{6}}+\sqrt{49-2\times5\times 2\sqrt{6}}=\\\\
=\sqrt{25+24+2\times5\times2\sqrt{6}}+\sqrt{25+24-2\times5\times 2\sqrt{6}}=\\\\
=\sqrt{25+2\times5\times2\sqrt{6}+24}+\sqrt{25-2\times5\times 2\sqrt{6}+24}=\\\\
=\sqrt{5^2+2\times5\times2\sqrt{6}+(2\sqrt{6})^2}+\sqrt{5^2-2\times5\times2\sqrt{6}+(-2\sqrt{6})^2}=\\\\
=\sqrt{(5+2\sqrt{6})^2}+\sqrt{(5-2\sqrt{6})^2}=5+2\sqrt{6}+5-2\sqrt{6}=\boxed{10}[/tex]
