Solution:

Denote $x=\overline{L A R N}-\overline{A C A}$, $y=\overline{C Y P}+\overline{R U S}$ and $z=C^{Y^{P}} \cdot R^{U^{S}}$. It is obvious that $1823-898 \leq x \leq 9187-121$, $135+246 \leq y \leq 975+864$, that is $925 \leq x \leq 9075$ and $381 \leq y \leq 1839$, whence it follows that $\frac{925}{1839} \leq \frac{x}{y} \leq \frac{9075}{381}$, or $0.502\ldots \leq \frac{x}{y} \leq 23.81\ldots$ Since $\frac{x}{y}=z$ is an integer, it follows that $1 \leq \frac{x}{y} \leq 23$, hence $1 \leq C^{Y^{P}} \cdot R^{U^{S}} \leq 23$. So both values $C^{Y^{P}}$ and $R^{U^{S}}$ are $\leq 23$. From this and the fact that $2^{2^{3}}>23$ it follows that at least one of the symbols in the expression $C^{Y^{P}}$ and at least one of the symbols in the expression $R^{U^{S}}$ correspond to the digit $1$. This is impossible because of the assumption that all the symbols in the set $\{C, Y, P, R, U, S\}$ correspond to different digits.