Given $${x^{{1 \over {50}}}} = 12 \Rightarrow x = {12^{50}}$$

$${y^{{1 \over {50}}}} = 18 \Rightarrow y = {18^{50}}$$

$$12\equiv13$$ (Mod 25)

$$12^2\equiv19$$ (Mod 25)

$$12^3\equiv-3$$ (Mod 25)

$$12^9\equiv-2$$ (Mod 25)

$$12^{10}\equiv-1$$ (Mod 25)

$$12^{50}\equiv-1$$ (Mod 25) ..... (i)

Now

$$18\equiv7$$ (Mod 25)

$$18^2\equiv-1$$ (Mod 25)

$$18^{-50}\equiv-1$$ (Mod 25) ..... (ii)

$$\therefore$$ $$12^{50}+18^{50}\equiv-2$$ (Mod 25)

$$\equiv23$$ (Mod 25)

$$\therefore$$ Answer = 23