1) Is $\displaystyle \sqrt{1} = 1$ ?

2) And if so, then $\displaystyle 1 + \sqrt{x} = \sqrt{1} + \sqrt{x} = \sqrt{1 + x}$ ?

3) If 1 and 2 is true, then:

$\displaystyle (1 + \sqrt{x})^2 = (\sqrt{1 + x})^2$

But if you actually square $\displaystyle (1 + \sqrt{x})^2 $ and $\displaystyle (\sqrt{1 + x})^2$, you'll find out that they are different:

$\displaystyle (1 + \sqrt{x})^2 = (1 + \sqrt{x})(1 + \sqrt{x}) = 1 + 2\sqrt{x} + x$

$\displaystyle (\sqrt{1 + x})^2 = 1 + x$

Why is that?