a: \(\frac{1}{\sqrt[3]{4-\sqrt{15}}}=\frac{\sqrt[3]{4+\sqrt{15}}}{\sqrt[3]{\left(4+\sqrt{15}\right)\left(4-\sqrt{15}\right)}}=\sqrt[3]{4+\sqrt{15}}\)
Đặt \(a=\sqrt[3]{4+\sqrt{15}};b=\sqrt[3]{4-\sqrt{15}}\)
=>\(a^3=4+\sqrt{15};b^3=4-\sqrt{15}\)
=>\(a^3+b^3=8;ab=1\)
x=a+b
=>\(x^3=\left(a+b\right)^3=a^3+b^3+3ab\left(a+b\right)\)
=>x^3=8+3x
=>\(x^3-3x-8=0\)
=>\(A=x^3-3x+2020=8+2020=2028\)
b: Đặt \(u=\sqrt[3]{\frac{25 + \sqrt{621}}{2}};v=\sqrt[3]{\frac{25 - \sqrt{621}}{2}}\)
=>\(y=\frac13\left(1-u-v\right)\)
=>1-u-v=3y
=>u+v=1-3y
\(u^3 + v^3 = \frac{25 + \sqrt{621}}{2} + \frac{25 - \sqrt{621}}{2} = \frac{50}{2} = 25\)
\(u \cdot v = \sqrt[3]{\frac{(25 + \sqrt{621})(25 - \sqrt{621})}{4}} = \sqrt[3]{\frac{625 - 621}{4}} = \sqrt[3]{\frac{4}{4}} = 1\)
\(\left(u+v\right)^3=u^3+v^3+3uv\left(u+v\right)\)
=>\(\left(1-3y\right)^3=25+3\cdot1\cdot\left(1-3y\right)\)
=>\(1 - 9y + 27y^2 - 27y^3 = 25 + 3 - 9y\)
=>\(1 + 27y^2 - 27y^3 = 28\)
=>\(-27y^3+27y^2=27\)
=>\(-y^3+y^2=1\)
B=2y^3-2y^2+2000
=2(y^3-y^2)+2000
=-2+2000
=1998
