Question

radical din 10+4√6 - radical din 10-4√6 sa se arate ca este nr nat

Answer 1

 

[tex]\displaystyle\\ \sqrt{10+4\sqrt{6}}-\sqrt{10-4\sqrt{6}}=\\\\ =\sqrt{10+2\times2\times\sqrt{6}}-\sqrt{10-2\times2\times\sqrt{6}}=\\\\ =\sqrt{6+4+2\times2\times\sqrt{6}}-\sqrt{6+4-2\times2\times\sqrt{6}}=\\\\ =\sqrt{6+2\times2\times\sqrt{6}+4}-\sqrt{6-2\times2\times\sqrt{6}+4}=\\\\ =\sqrt{(\sqrt{6})^2+2\times2\times\sqrt{6}+2^2}-\sqrt{(\sqrt{6})^2-2\times2\times\sqrt{6}+2^2}=\\\\ =\sqrt{(\sqrt{6}+2)^2}-\sqrt{(\sqrt{6}-2)^2}=\\\\ =\sqrt{6}+2-(\sqrt{6}-2)=\\\\ =\sqrt{6}+2-\sqrt{6}+2=2+2=\boxed{\bf 4\in N}[/tex]