Confusion about ##\sqrt{x^2}= \left| x \right|##

[RChristenk]

Homework Statement

Confusion about $\sqrt{x^2}= \left| x \right|$

Relevant Equations

Absolute value concept

By definition, $\sqrt{x^2}= \left| x \right|$.

For positive $x$, such as $4$, it is quite straightforward: $\sqrt{4^2}=\sqrt{16}=4$.

For negative values, I am more confused: $\sqrt{(-4)^2}=\sqrt{16}=4$. The answer will always be positive, even if you put in a negative value. So why is there a need for the absolute value on the right side? Wouldn't $\sqrt{x^2}=x$ always work? Because after $x^2$ inside the square root, the value will always be positive.

I am guessing the absolute value is to prevent something like $\sqrt{(-4)^2} = -4$, but if you just focus on the operation on the left hand side of the equal sign, it is not possible to ever end up with a negative value on the right hand side. So why the need for the absolute value? Thanks.

 

[Mark44]

For negative values, I am more confused: $\sqrt{(-4)^2}=\sqrt{16}=4$. The answer will always be positive, even if you put in a negative value. So why is there a need for the absolute value on the right side? Wouldn't $\sqrt{x^2}=x$ always work? Because after $x^2$ inside the square root, the value will always be positive.

Because $\sqrt{(-4)^2}=4 \ne -4$. So this is an example of why $\sqrt{x^2} \ne x$.