Answer:
There are two situations involving rational exponents or radicals that will never result in a negative real solution:
Even-indexed roots: If we take the square root, fourth root, sixth root, etc. of a non-negative real number, the result will always be non-negative. For example, the square root of 9 is 3, and the fourth root of 16 is 2, both of which are non-negative. This is because even-indexed roots always produce a non-negative result, regardless of the sign of the original number.
Exponents with even denominators: If we raise a non-negative real number to an exponent with an even denominator, the result will always be non-negative. For example, (4^2/4) is equal to 4, which is non-negative. This is because any negative base raised to an even power results in a positive number, and any positive base raised to an even power also results in a positive number. Therefore, any exponent with an even denominator will always produce a non-negative result, regardless of the sign of the original number.