$81^{81.25} = 3\cdot 81^{81}$
$82^{81} =81^{81} \left(\dfrac{1+81}{81}\right)^{81}$
$\lim_{n \to \infty} ~\left(\dfrac{1+n}{n}\right)^n = e$
and this limit is approached strictly from below.
Thus $\left(\dfrac{1+81}{81}\right)^{81} < e < 3$
So comparing the two expressions we have
$3 \cdot 81^{81} > \left(\dfrac{1+81}{81}\right)^{81} \cdot 81^{81}$
and thus $81^{81.25} > 82^{81}$