Explain why rational exponents are not defined when the denominator of the exponent in lowest terms is even and the base is negative

The correct answer is:

we cannot take an even root of a negative number in the real number system.

Explanation:

A rational exponent is another way of writing a root. The denominator of the rational exponent will be the root, and the numerator will be the power of the number under the root. If the denominator is even, this means we are taking an even root, such as square root, 4th root, etc.

If the base is negative, this means we want an even root of a negative number.

Any even root can be broken down into a series of square roots, done one after another.

However, we cannot take the square root of a negative number using real numbers; this is because a square root is a number multiplied by itself to get the answer; a negative times a negative is a positive, as is a positive times a positive, so there is no way to multiply the same number twice and get a negative.

We cannot take an even root of a negative number in the real number system.