Originally Posted by

**kwah** I'm really confused where to begin with replying to this ..

first off, i don't agree that $\displaystyle \sqrt{x^2} = |x|$

regardless of the value of $\displaystyle x$, positive or negative, it results in $\displaystyle \sqrt{positive number}$, yeah?

roots can have a positive OR negative value, therefore $\displaystyle \sqrt{x^2}$ does not necessarily equate to $\displaystyle |x|$

If, as I stated in my last post, the $\displaystyle ^2$ is OUTSIDE the root, then $\displaystyle x$ is forced to become positive for two reasons..

$\displaystyle \sqrt{x}^2 = (\sqrt{x})^2 = |x|$ is true (note: the $\displaystyle ^2$ outside the root)

for the reasons we have both stated, you cannot root negative numbers therefore x is required to be positive..

AND, squaring it afterwards would force it to become positive regardless..

the only other confusions / point I have to make about this is

$\displaystyle \sqrt{x^2} = (x^2)^0.5 = x^1$

$\displaystyle \sqrt{(-10)^2} = ((-10)^2)^0.5 = (-10)^1$ (negative)