# Abs(x) = sqrt(x^2) Proof

1. Dec 31, 2011

### basil32

1. The problem statement, all variables and given/known data
Prove that |x| = sqrt(x^2)

3. The attempt at a solution
I've written two proofs but I don't know if they can be justified as real proofs or whether they are valid or not.
Proof 1:
$\surd x^{2} = \surd \vert x \vert ^{2} = \vert x \vert$

Proof 2:
First Case ) Suppose $x \geq 0$ then $\surd x^{2} = x = \vert x \vert$
Second Case ) Suppose $x < 0$ then $\surd x^{2} = -x$ where $-x > 0$ therefore $-x = \vert x \vert$

2. Dec 31, 2011

### gb7nash

Let's look at the definition of the square root:

If a2 = b and a ≥ 0, then a = √b. Now look at your problem. What two things do we have to prove?

3. Dec 31, 2011

### basil32

That $x^{2} \geq 0$ and $\vert x \vert \geq 0$ ?

4. Dec 31, 2011

### gb7nash

The two bolded things are what you want to prove. Once you have those, then the conclusion follows. Before you do anything, what is a in your problem? What is b? Once you have a and b, what is the first thing you need to prove?

5. Dec 31, 2011

### basil32

$a = \vert x \vert$ and $b = x^{2}$

$a^{2} = \vert x \vert ^{2} = x ^ {2} = b$
$a = \vert x \vert$ which is nonnegative. correct?

6. Dec 31, 2011

Correct.