## Why does \(\sqrt{x^{2}}=|x|\) and not simply x?

Answer:

Explanation:

Let’s take a quick example and see why \(\sqrt{x^{2}}=|x|\) and not simply \(x\).

Let’s set \(x=3,-3\) and put them into the \(\sqrt{x^{2}}\) term:

\(x=3 \Rightarrow \sqrt{x^{2}} \Rightarrow \sqrt{3^{2}}=\sqrt{9}=?\)

\(x=-3 \Rightarrow \sqrt{x^{2}} \Rightarrow \sqrt{(-3)^{2}}=\sqrt{9}=?\)

What should be in the question mark spaces? In both these cases, we can end up with the original value of \(x\) but we can also end up with the opposite sign, or \(-x\) Our general formula needs to reflect that.

Since we can get either the same sign or the opposite sign from this operation, we need to show it in our general formula. One way to do it would be to say something like \(\sqrt{x^{2}}=\pm x\). However, this could become messy. And so the absolute value is used to express that the value of \(x\) could be positive or it could be negative.