Question

Calculati va rog ecuatia:√(49+20√6)+√(49-20√6)

Answer 1

   
[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]