Can x be equal to the square root of its own absolute value?

[basil32]

Homework Statement

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

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.