Square root of a number squared is equal to the absolute value of that number

Question

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.

Answer 1

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."

Answer 2

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.