In 1748, Leonhard Euler introduced variable exponents, and, implicitly, non-integer exponents by writing:

The exponentiation operation with integer exponents may be defined directly from elementary arithmetic operations.

This definition is the only possible that allows extending the formula

The powers of a sum can normally be computed from the powers of the summands by the binomial formula

The limit of a sequence of powers of a number greater than one diverges; in other words, the sequence grows without bound:

If the exponentiated number varies while tending to 1 as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is

In all cases, the complex logarithm is used to define complex exponentiation as

Some identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined as single-valued functions. For example:

Then compute the following terms in order, reading Horner's rule from right to left.

Still others only provide exponentiation as part of standard libraries: