How does a fractional power operate in mathematics?

A fractional power in mathematics operates by representing both a power and a root simultaneously. Specifically, when you have a base raised to a fractional exponent, the numerator of the fraction indicates the power to which the base is raised, while the denominator indicates the root that is taken of the base.

For instance, if you have ( a^{\frac{m}{n}} ), this can be understood as ( (a^m)^{\frac{1}{n}} ) or equivalently ( \sqrt[n]{a^m} ). This interpretation shows that the denominator specifies the degree of the root — meaning you are taking the n-th root of the base. If the numerator was 1, it simplifies to the root of the base itself, illustrating that the fractional power combines both exponentiation and root extraction in one expression.

Other choices do not accurately describe the operation of fractional powers. For example, raising the base to a whole number does not involve fractions at all since it pertains to integers alone. Simplifying the expression and converting the base to a fraction also do not encapsulate the fundamental nature of fractional powers, as these actions pertain to different mathematical processes.