If $$p + \frac{1}{p} = 112,$$   find the value of $${\left( {p - 112} \right)^{15}} + \frac{1}{{{p^{15}}}} = ?$$

$$\eqalign{ & p + \frac{1}{p} = 112,\,\,{\left( {p - 112} \right)^{15}} + \frac{1}{{{p^{15}}}} = ? \cr & p - 112 = - \frac{1}{p} \cr & {\left( {p - 112} \right)^{15}} + \frac{1}{{{p^{15}}}} \cr & = {\left( { - \frac{1}{p}} \right)^{15}} + \frac{1}{{{p^{15}}}} \cr & = - \frac{1}{{{p^{15}}}} + \frac{1}{{{p^{15}}}} \cr & = 0 \cr} $$