[tex]\displaystyle\\
\text{Testul 2, Ex. 6.}\\\\
\sqrt{10+4 \sqrt{6}}- \sqrt{10-4 \sqrt{6}}=?\\\\
\text{Explicatie:}\\
\text{Transformam expresia de sub radicalul mare intr-o }\\
\text{expresie la puterea a 2-a de forma: } \Big(a + b \sqrt{c} \Big)^2\\\\
\text{Descompunem in 3 factori termenul care contine radical: } 4 \sqrt{6}.\\
\text{Descompunerea nu este unica, dar o singura varianta de de}\\
\text{descompunere este utila.}\\
\text{Primul factor este obligatoriu 2, dar nu-l folosim imediat.}\\
[/tex]
[tex]\displaystyle\\
\text{Ceilalti 2 factori ii folosim in felul urmator.}\\
\text{Calculam suma patratelor lor. }\\
\text{Aceasta suma trebuie sa fie egala cu primul termen }\\
\text{de sub radicalul mare, 10 in cazul nostru.}\\
\text{Daca nu este egala refacem descompunerea in factori.}\\
\text{Daca este egala, am terminat ce-a fost mai greu.}[/tex]
[tex]\displaystyle\\
\text{Rezolvare:}\\\\
\sqrt{10+4\sqrt{6}}-\sqrt{10-4\sqrt{6}}=?\\\\
4\sqrt{6}=2\times2\times\sqrt{6}\\
2^2+\Big(\sqrt{6}\Big)^2=4+6=10\Longrightarrow OK!\\\\
[/tex]
[tex]\displaystyle\\
\sqrt{10+4\sqrt{6}}-\sqrt{10-4\sqrt{6}}=\\\\
=\sqrt{6+4+4\sqrt{6}}-\sqrt{6+4-4\sqrt{6}}=\\\\
=\sqrt{6+4\sqrt{6}+4}-\sqrt{6-4\sqrt{6}+4}=\\\\
=\sqrt{\Big(\sqrt{6}\Big)^2+2\times\sqrt{6}\times2+2^2}-\sqrt{\Big(\sqrt{6}\Big)^2-2\times\sqrt{6}\times 2+2^2}=\\\\
=\sqrt{\Big(\sqrt{6}+2\Big)^2}-\sqrt{\Big(\sqrt{6}-2\Big)^2}=\\\\
=\sqrt{6}+2-(\sqrt{6}-2)=\sqrt{6}+2-\sqrt{6}+2=2+2=\boxed{\bf 4\in N}[/tex]
