In this paper, the notion of quaternionic Balakrishnan operator \(J^{\alpha }\) with the quaternionic power ( \(\alpha \in {\mathbb {H}}\) and \(\textrm{Re}(\alpha )>0\) ) is introduced via quaternionic non-negative operator T and the inclusion relations of the domains and ranges of these two types of operators are demonstrated. The unique unified integral representation of \(J^{\alpha }\) is obtained through the slice Cauchy kernels and the slice regularity of the exponent mapping is proved. Further, the limits of \(J^{\alpha }x\) as \(\alpha \rightarrow 0\) and \(\alpha \rightarrow 1\) are investigated under some fixed spherical sectors. By obtaining moment inequality, we prove that \(J^{\alpha }\) is a closable operator and introduce the power with base T and exponent \(\alpha \) as the operator \(\overline{J^{\alpha }}\) , then the integral representation of \(\overline{J^{\alpha }}\) is given by the limit of S-resolvent operator of \(-T\) . Besides, the related integral expressions of \(J^{\alpha }\) are established via quaternionic semigroup when \(-T\) is the infinitesimal generator of an equibounded strongly continuous quaternionic semigroup. It is crucial to note that the fractional quaternionic operator set \(\{\overline{J_{T}^{\alpha }}:\alpha \in {\mathbb {H}}^{+}\}\) has a nice semigroup property under the \(*\) -product we introduced under the noncommutative setting. In addition, for the space consisting of right linear bounded quaternionic operators with a Schauder basis, we also obtained the semigroup property for \(J^{\alpha }\) through introducing the \(\star \) -product.
