Tính giá trị biểu thức: B=căn bậc 3(85+62căn7)+ căn bậc 3(85−62căn7

* Hướng dẫn giải

$B=\sqrt[3]{85+62\sqrt{7}}+\sqrt[3]{85-62\sqrt{7}}$

Đặt $a=\sqrt[3]{85+62\sqrt{7}};b=\sqrt[3]{85-62\sqrt{7}}=>a+b=B$

Mặt khác $$\begin{gathered} a^3+b^3=(85+62\sqrt{7})+(85-62\sqrt{7})=170 \\ ab=\sqrt[3]{85+62\sqrt{7}}\sqrt[3]{85-62\sqrt{7}}=\sqrt[3]{{85}^2-{\left(62\sqrt{7}\right)}^2}=\sqrt[3]{-19683}=-27 \end{gathered}$$

Ta có:

 $\begin{array}{l}{B}^3={\left(a+b\right)}^3=a^3+b^3+3ab(a+b)\\ =170-\mathrm{3.27.}B\\ =>B^3+81B-170=0\\ =>(B-2)\left(\underset{>0}{\underbrace{B^2+2B+85}}\right)=0\\ =>B=2\end{array}$