Question

Testul 2 ex. 6...nu reușesc

Answer 1

   
[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]