Question

why is the expression x1/2 is undefined when x is less than 0?

Answer 1

Use the definition of a square root.

Explanation:

Observe that `x^(1/2) = sqrt(x)`.

The value of `sqrt(x)` is the non-negative real number whose square is `x`.
Let `c = sqrt(x)`, just to give it a name.
If x = 0 then c = 0.

Otherwise `c^2 = x`, and `c ne 0`.

If c is a positive real number, then `c^2 = x` is a positive number times a positive number, which is positive. So `x > 0`.

If c is a negative real number, then `c^2` is a negative number times a negative number, which is positive. So `x > 0`.

It is impossible for the square of a real number to be negative.
Therefore, it is impossible for x to be negative.