21. Find the greatest integer less than or equal to $(2+\sqrt{3})^{3}$.
21. Answer: 51
Using binomial theorem it is easy to see that
$$
(2+\sqrt{3})^{3}+(2-\sqrt{3})^{3}=52,
$$
and that $0<2-\sqrt{3}<1$, or $0<(2-\sqrt{3})^{3}<1$ so that we have $51<(2+\sqrt{3})^{3}<52$.