Final answer:
The correct rule for roots with even indexes is that they can be neither exclusively negative nor positive; the principal root of a real number is always non-negative because you cannot take an even root of a negative number and get a real result.
Step-by-step explanation:
The correct rule for roots with even indexes (2, 4, 6, etc.) is they can be neither negative nor positive exclusively. Instead, the principal (or primary) root is always non-negative for a real number because you cannot take an even root of a negative number and get a real number result.
Let's consider some examples for clarification:
- For the square root (an index of 2), √4 = 2. The result is a positive number.
- For the fourth root (an index of 4), √√16 = −4 = 2. Again, the result is positive.
- For any negative number, like -16, the fourth root would not be a real number. There are no real fourth roots of negative numbers.
roots with even indexes, the statement that they are 'always irrational' (Option A) is incorrect because roots can be rational or irrational depending on the number. They are 'not always negative' (Option B), as shown in the examples above. They are 'not always integers' (Option D) either; for example, the square root of 2 is irrational. Thus, the correct answer to the question is 'They can be negative or positive' (Option C), with the understanding that in the set of real numbers, they can only be non-negative.