What mathematical operation is represented by the expression x^a/b?

The expression ( x^{a/b} ) represents the mathematical concept of applying a power to another power, specifically in the context of roots and exponents. When we see an exponent expressed as a fraction, such as ( a/b ), it denotes that we are dealing with both exponentiation and root extraction in the same expression.

In this case, ( x^{a/b} ) can be interpreted as taking the ( b )-th root of ( x^a ). This means that you first raise ( x ) to the power ( a ) and then take the ( b )-th root of that result. This dual operation of exponentiation and root calculation aligns closely with the principles of fractional exponents in mathematics.

This understanding of fractional exponents adds crucial insight into how powers and roots are interconnected, making interpretation intuitive within the broader context of algebra. Thus, the correct answer highlights the relationship between exponentiation and the concept of roots, categorizing the operation performed in the expression.