Final answer:
The 'index' of 2^-3/5 refers to the exponent (-3/5) and implies taking the reciprocal of the cube of the fifth root of 2. The -3 signifies the reciprocal, while 1/5 signifies the fifth root.
Step-by-step explanation:
The question asks about the index of the expression 2^-3/5. The term 'index' in this context is a bit unclear but could refer to the exponent itself or the root when dealing with rational exponents. Since the base number here is 2 and it is raised to the negative fractional power of -3/5, this could imply taking the fifth root of 2 and then raising it to the power -3, which means taking the reciprocal of 2^3 after extracting the fifth root.
Rational exponents can be solved by applying the rules of exponents. For instance, 2^-3/5 could first be considered as 2^(1/5) (which is the fifth root of 2), then raised to the -3 power, resulting in the reciprocal of the cube of the fifth root of 2 because a negative exponent indicates a reciprocal. Remember that when we divide powers with the same base, we subtract the exponents as highlighted by numerous mathematical operations provided in the references. Therefore, we can say that the index is both -3, which indicates the reciprocal, and 1/5, which indicates the extraction of the fifth root of 2.