Question

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

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Answer 1

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

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

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

\(=\dfrac{\left(3\sqrt{2}+\sqrt{3}+\sqrt{6}+4\right)\left(\sqrt{2}+\sqrt{3}-2\right)}{1+2\sqrt{6}}\)

\(=\dfrac{6+3\sqrt{6}+6\sqrt{2}+\sqrt{6}+3-2\sqrt{3}+\sqrt{12}+\sqrt{18}-2\sqrt{6}+4\sqrt{2}+4\sqrt{3}-8}{1+2\sqrt{6}}\)

\(=\dfrac{6+3\sqrt{6}-6\sqrt{2}+\sqrt{6}+3-2\sqrt{3}+2\sqrt{3}+\sqrt{18}+2\sqrt{6}+4\sqrt{2}+4\sqrt{3}-8}{1+2\sqrt{6}}\)

\(=\dfrac{6+3\sqrt{6}-6\sqrt{2}+\sqrt{6}+3+3\sqrt{2}-2\sqrt{6}+4\sqrt{2}+4\sqrt{3}-8}{1+2\sqrt{6}}\)

\(=\dfrac{1+2\sqrt{6}+\sqrt{2}+4\sqrt{3}}{1+2\sqrt{6}}\)

\(=1+\sqrt{2}\)

Answer 2

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

\(C=\dfrac{\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{2}\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)}{\sqrt{2}+\sqrt{3}+\sqrt{4}}=\dfrac{\left(1+\sqrt{2}\right)\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)

\(C=1+\sqrt{2}\)