Exponentiation
When it is a positive integer, the exponent indicates how many copies of the base are multiplied together. For example, . The base appears times in the repeated multiplication, because the exponent is . Here, is the "base", is the "exponent", and is the "power" or, more specifically, "3 raised to the 5th power".
Exponential family
The derivation is a simple variational calculation using Lagrange multipliers. Normalization is imposed by letting "T" = 1 be one of the constraints. The natural parameters of the distribution are the Lagrange multipliers, and the normalization factor is the Lagrange multiplier associated to "T".