14. (40 points) Determine the sign of the sum

$$
(\sqrt{2021}-\sqrt{2020})^{3}+(\sqrt{2020}-\sqrt{2019})^{3}+(\sqrt{2019}-\sqrt{2021})^{3}
$$

without raising the addends to the third power.
Answer: The expression is negative.

Solution. If $a+b+c=0$, then $a^{3}+b^{3}+c^{3}=3 a b c$, since

$$
(a+b+c)^{3}=a^{3}+b^{3}+c^{3}-3 a b c+(a b+b c+a c)(a+b+c)
$$

We have $a=\sqrt{2021}-\sqrt{2020}>0, b=\sqrt{2020}-\sqrt{2019}>0, c=\sqrt{2019}-\sqrt{2021}<0$.