Question

\(\dfrac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)

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Answer 1

\(\dfrac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\) 

= \(\dfrac{\left(\sqrt{2}+\sqrt{3}+2\right)+\left(\sqrt{6}+\sqrt{8}+2\right)}{\sqrt{2}+\sqrt{3}+2}\)

= \(\dfrac{\left(\sqrt{2}+\sqrt{3}+2\right)+\sqrt{2}\left(\sqrt{3}+2+\sqrt{2}\right)}{\left(\sqrt{2}+\sqrt{3}+2\right)}\)

= \(\dfrac{\left(\sqrt{2}+\sqrt{3}+2\right)\times\left(1+\sqrt{2}\right)}{\left(\sqrt{2}+\sqrt{3}+2\right)}\)

= 1 + \(\sqrt{2}\)