Question

Sper sa ma ajutati :)

Answer 1

 

$\displaystyle\bf\\49)\\\\\frac{\sqrt{6}}{\sqrt{8}+\sqrt{10}}+\frac{\sqrt{8}}{\sqrt{6}+\sqrt{10}}+\frac{\sqrt{10}}{\sqrt{6}+\sqrt{8}}=\\\\\\=\frac{\sqrt{6}}{\sqrt{10}+\sqrt{8}}+\frac{\sqrt{8}}{\sqrt{10}+\sqrt{6}}+\frac{\sqrt{10}}{\sqrt{8}+\sqrt{6}}=\\\\\\\textbf{Rationalizam numitorul fiecarei fractie.}\\\\\\=\frac{\sqrt{6}(\sqrt{10}-\sqrt{8})}{10-8}+\frac{\sqrt{8}(\sqrt{10}-\sqrt{6})}{10-6}+\frac{\sqrt{10}(\sqrt{8}-\sqrt{6})}{8-6}=$

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$\displaystyle\bf\\=\frac{\sqrt{60}-\sqrt{48}}{2}+\frac{\sqrt{80}-\sqrt{48}}{4}+\frac{\sqrt{80}-\sqrt{60}}{2}=\\\\\\=\frac{2\sqrt{15}-4\sqrt{3}}{2}+\frac{4\sqrt{5}-4\sqrt{3}}{4}+\frac{4\sqrt{5}-2\sqrt{15}}{2}=\\\\\\=\sqrt{15}-2\sqrt{3}+\sqrt{5}-\sqrt{3}+2\sqrt{5}-\sqrt{15}=\\\\=\Big(\sqrt{15}-\sqrt{15}\Big)+\Big(\sqrt{5}+2\sqrt{5}\Big)-2\sqrt{3}-\sqrt{3}=\\\\=0+3\sqrt{5}-3\sqrt{3}=\\\\=3\Big(\sqrt{5}-\sqrt{3}\Big)\approx3\Big(2,236-1,732\Big)\approx3\times0,504 \approx1,512<2$