\(37-30\sqrt{3}=1^3-3.1^2.2\sqrt{3}+3.1.\left(2\sqrt{3}\right)^2-\left(2\sqrt{3}\right)^3=\left(1-2\sqrt{3}\right)^3\)
Thay vào A ta được:
\(A=\left(3+2\sqrt{3}\right)\sqrt{33-12\sqrt{5-\sqrt[3]{\left(1-2\sqrt{3}\right)^3}}}\)
\(=\left(3+2\sqrt{3}\right)\sqrt{33-12\sqrt{5-1+2\sqrt{3}}}\)
\(=\left(3+2\sqrt{3}\right)\sqrt{33-12\sqrt{3+2\sqrt{3}+1}}\)
\(=\left(3+2\sqrt{3}\right)\sqrt{33-12\sqrt{\left(\sqrt{3}+1\right)^2}}\)
\(=\left(3+2\sqrt{3}\right)\sqrt{33-12\left(\sqrt{3}+1\right)}\)
\(=\left(3+2\sqrt{3}\right)\sqrt{33-12\sqrt{3}-12}\)
\(=\left(3+2\sqrt{3}\right)\sqrt{12-2.\left(2\sqrt{3}\right).3+9}\)
\(=\left(3+2\sqrt{3}\right)\sqrt{\left(2\sqrt{3}-3\right)^2}\)
\(=\left(3+2\sqrt{3}\right)\left(2\sqrt{3}-3\right)\) (vì \(2\sqrt{3}>3\))
\(=12-9=3\)
