27. Let $m \neq n$ be two real numbers such that $m^{2}=n+2$ and $n^{2}=m+2$. Find the value of $4 m n-m^{3}-n^{3}$
27. Ans: 0 .
Since $m \neq n$ and $m^{2}-n^{2}=(n-m)$, we get $m+n=-1$. Thus $m^{2}+n^{2}=(m+n)+4=3$ and
$$
\begin{array}{l}
m n=\frac{1}{2}\left[(m+n)^{2}-m^{2}-n^{2}\right]=-1 \text {. Thus } 4 m n-m^{3}-n^{3}=4 m n-(m+n)\left(m^{2}+n^{2}-m n\right)= \\
-4-(-1)(3-(-1))=0 .
\end{array}
$$