On power of singular cardinals.

Using elementary methods we find bounds for the function 2* α for KQ, = a.Using only ZFC without additional assumptions, when e.g., K α is strong limit of uncountable confinality:(1) If there is no weakly inaccessible below K α , then there is no such cardinal below 2 K «. (2) If K α is the first cardinal such that λ = K λ with cfλ = X U then 2*<* < K when K is the first cardinal such that K = K κ with cofinality (2 2 *') + .We shall also reprove some of Galvin and HajnaFs results.We do not require any knowledge of earlier results on the subject.* > K α (Cantor's theorem).