Question

Cho \(a=\sqrt{3+\sqrt{5+2\sqrt{3}}}+\sqrt{3-\sqrt{5+2\sqrt{3}}}\)

Tính \(M=a^4-3a^3+2a+2015\)

Answer 1

Bình phương a ta được

\(a^2=3+3+\sqrt{5+2\sqrt{3}}-\sqrt{5+2\sqrt{3}}+2\sqrt{\left(3+\sqrt{5+2\sqrt{3}}\right)\left(3-\sqrt{5+2\sqrt{3}}\right)}\)

\(a^2=6+2\sqrt{9-3\sqrt{5+2\sqrt{3}}+3\sqrt{5+2\sqrt{3}}-5-2\sqrt{3}}\)

\(a^2=6+2\sqrt{9-5-2\sqrt{3}}\Rightarrow a^2=6+2\sqrt{4-2\sqrt{3}}\Rightarrow a^2=6+2\sqrt{3+1-2.1.\sqrt{3}}\)\(a^2=6+2\sqrt{\left(\sqrt{3}-1\right)^2}\Rightarrow a^2=6+2\sqrt{3}-2=4+2\sqrt{3}=3+1+2.1.\sqrt{3}=\left(\sqrt{3}+1\right)^2\Rightarrow a=\sqrt{3}+1\)

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