Question

Tính A=\(\sqrt[3]{5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}\)

Answer 1

đặt\(a=\sqrt[3]{5+2\sqrt{13}}\\ b=\sqrt[3]{5-\sqrt{13}}\)

ta có \(A^3=a^3+3ab\left(a+b\right)+b^3=5+2\sqrt{13}+5-2\sqrt{13}\\ \)

<=>\(A^3=10+3\sqrt[3]{\left(5+2\sqrt{13}\right)\left(5-2\sqrt{13}\right)}\cdot A\)

<=>\(A^3=10-9A\)

<=>\(A^3+9A-10=0\)\(\)

<=>\(A^3+10A-A-10=0\)

<=>\(A\left(A^2-1\right)+10\left(A-1\right)=0\)

<=>\(\left(A-1\right)\left(A^2+A+10\right)=0\)

Vì \(A^2+A+10>0\left(\forall A\right)\)

\(=>A-1=0\\ A=1\)