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

1 Answer
Mar 1, 2018

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.