Possible Duplicate:
Significance of $\displaystyle\sqrt[n]{a^n} $?
The square root of a number squared is equal to the absolute value of that number. Why is $\sqrt{x^2} = |x|$? Why not just $x$? Please give me a reason and also help me prove it.
Possible Duplicate:
Significance of $\displaystyle\sqrt[n]{a^n} $?
The square root of a number squared is equal to the absolute value of that number. Why is $\sqrt{x^2} = |x|$? Why not just $x$? Please give me a reason and also help me prove it.
You are referring to the principal square root. From Wikipedia: "Although the principal square root of a positive number is only one of its two square roots, the designation 'the square root' is often used to refer to the principal square root."
By definition the square root of a nonnegative real number $y$ is the unique real number $z$ for which $z\geq 0$ and $z^2=y$. Consequently, to prove that $\sqrt{x^2}=|x|$ it is enough to show that $|x|\geq 0$ and $|x|^2=x^2$ but these two facts are very easy to verify.