Question

What is `(sqrt(2x+4))^2`?

Answer 1

`(sqrt(2x+4))^2 = 2x+4` for all `x in RR` or for all `x in [-2, oo)` if you only consider `sqrt` as a real valued function.

Explanation:

Note that if `x < -2` then `2x+4 < 0` and `sqrt(2x+4)` has a complex (pure imaginary) value, but its square will still be `2x+4`.

Essentially, `(sqrt(z))^2 = z` by definition. If the square root exists, then it is a value whose square gives you back the original value.

Interestingly, `sqrt((2x+4)^2) = abs(2x+4)` not `2x+4`