23 Evaluate $\sqrt[3]{77-20 \sqrt{13}}+\sqrt[3]{77+20 \sqrt{13}}$.
23 Answer: (7)
Let $X=\sqrt[3]{77-20 \sqrt{13}}, Y=\sqrt[3]{77+20 \sqrt{13}}$ and $A=X+Y$.
Since $X^{3}+Y^{3}=154, X Y=\sqrt[3]{77^{2}-20^{2} \times 13}=9, A^{3}=(X+Y)^{3}=X^{3}+Y^{3}+3 X Y(X+Y)$ $\Rightarrow A^{3}=154+27 A \Rightarrow A^{3}-27 A-154=0 \Rightarrow(A-7)\left(A^{2}+7 A+22\right)=0 \Rightarrow A=7$