Blackbox wrote:

Going thru the beginning chapters from GMAT Math book v3, it reads:

\(\sqrt{x^2}\) = |x| and

when x<=0 then \(\sqrt{x^2}\) = -x and

when x>= 0 then \(\sqrt{x^2}\) = x.

My Math is rusty but don't you have to square both sides in order to remove the modulus? Pardon my ignorance on the topic. Maybe someone could help me understand the concept?

p.s: I did search for this question before I posted but the search yielded no results due to too many common parameters provided.

MUST KNOW: \(\sqrt{x^2}=|x|\):The point here is that since

square root function cannot give negative result then \(\sqrt{some \ expression}\geq{0}\).

So \(\sqrt{x^2}\geq{0}\). But what does \(\sqrt{x^2}\) equal to?

Let's consider following examples:

If \(x=5\) --> \(\sqrt{x^2}=\sqrt{25}=5=x=positive\);

If \(x=-5\) --> \(\sqrt{x^2}=\sqrt{25}=5=-x=positive\).

So we got that:

\(\sqrt{x^2}=x\), if \(x\geq{0}\);

\(\sqrt{x^2}=-x\), if \(x<0\).

What function does exactly the same thing? The absolute value function: \(|x|=x\), if \(x\geq{0}\) and \(|x|=-x\), if \(x<0\). That is why \(\sqrt{x^2}=|x|\).

Hope it's clear.

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