In this paper, arities of relations are considered to be arbitrary sets. We introduce and study a new operation of product of universal relational systems, which lies between their direct product and the direct product of their reflexive hulls. For the new operation of product and the direct sum of universal relational systems, the validity of the distributive law is shown. Moreover, we define a new power of universal relational systems by combining their direct power and structural power. Then, all the three powers are discussed. It is shown that the introduced power of universal relational systems satisfies the first exponential law with respect to the combined product. Further, we show that the weak forms of the second and third exponential laws for each of the three powers of universal relational systems with respect to the new operation of product are satisfied.