abs(x) = sqrt(x^2) Proof

[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

[gb7nash]

Prove that |x| = sqrt(x^2)


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?

[basil32]

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?

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

[gb7nash]

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

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?

[basil32]

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?

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?

[gb7nash]

Correct.