Question

√((6−4×√(2))+√((6+4×√(2))
Cerinta: aratati ca urmatoarele numere sunt rationale

Answer 1

   
[tex]\displaystyle\\ \sqrt{6-4\sqrt2}+\sqrt{6+4\sqrt2} = \\\\ =\sqrt{4+2-2\times 2\times\sqrt2}+\sqrt{4+2+2\times 2\times\sqrt2} = \\\\ =\sqrt{4-2\times 2\times\sqrt2 +2}+\sqrt{4+2\times 2\times\sqrt2+2} = \\\\ =\sqrt{2^2-2\times 2\times\sqrt2 +(-\sqrt2)^2}+\sqrt{2^2+2\times 2\times\sqrt2+(\sqrt2)^2} = \\\\ =\sqrt{(2-\sqrt2)^2}+\sqrt{(2+\sqrt2)^2} = \\\\ = 2-\underline{\sqrt2} + 2+\underline{\sqrt2} = 2 + 2 = \boxed{4 \in \mathbb{Q}} [/tex]