Question

Tính F=(x³-12x-31)²⁰¹⁹

Với x=\(\sqrt{4+2\sqrt{3}}-\sqrt{4-2\sqrt{3}}\)

Answer 1

Ta có : \(x=\sqrt{4+2\sqrt{3}-\sqrt{4-2\sqrt{3}}}\)

=> \(x=\sqrt{3+2\sqrt{3}+1-\sqrt{3-2\sqrt{3}+1}}\)

=> \(x=\sqrt{\left(\sqrt{3}+\sqrt{1}\right)^2}-\sqrt{\left(\sqrt{3}-\sqrt{1}\right)}^2\)

=> \(x=|\sqrt{3}+\sqrt{1}|-|\sqrt{3}-\sqrt{1}|\)

=> \(x=\left(\sqrt{3}+\sqrt{1}\right)-\left(\sqrt{3}-\sqrt{1}\right)\)

=> \(x=\sqrt{3}+\sqrt{1}-\sqrt{3}+\sqrt{1}=\sqrt{1}+\sqrt{1}=1+1=2\)

Thay x = 2 vào biểu thức F ta được :

\(F=\left(2^3-12.2-31\right)^{2019}\)

=> \(F=\left(-47\right)^{2019}\)

Vậy F = (-49)^2019