In this article, we establish certain identities about the closedness of the range of several classes of possibly unbounded operators defined on a Hilbert space, such as self-adjoint, positive, normal, hyponormal, and quasinormal operators as well as their powers, and fractional powers when the latter applies. In particular, we show that an unbounded self-adjoint positive operator has closed range if and only if one (and all) of its fractional powers has closed range. A substantial part of the paper is devoted to additivity properties of operator ranges, again in the context of non-necessarily bounded operators. The paper is accompanied by many counterexamples that demonstrate the boundaries of possible assertions.