why is $\sqrt{x^2} \ne x$?

Question

If $\sqrt{3^2}= 3$, $\sqrt{2^2}\ne 2$.

why is $\sqrt{x^2}\ne x$?

Answer 1

$\sqrt{x^2} = x$ if $x \geq 0$. Now try a negative number, like $x = -5$. Then $$\sqrt{x^2} = \sqrt{(-5)^2} = \sqrt{25} = 5.$$ So when $x < 0$, it will become positive once we square; that is, $x^2 > 0$. Taking the square root of $x^2$ yields $-x > 0$.

In general, $\sqrt{x^2} = |x|$ for every real number $x$.

Answer 2

The short answer is that $x^{2}$ takes a negative number to a positive number, so $\sqrt{x^{2}}$ also takes a negative number to a positive number. Take as an example $-1$:

$$\sqrt{(-1)^{2}} = \sqrt{1} = 1$$