# what is the square root of 144

I am preparing for GRE and can anybody explain this to me

What is $\sqrt{144}$.

Why is the answer not $12 , -12.$

The calculator gives 12. does it mean -12 is incorrect?

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By definition, the square root must be non-negative. – TerranDrop Dec 11 '13 at 13:48
If you are solving something like $x^2=144$, make sure to use the absolute value sign when extracting the radical, i.e., $|x|=12\iff x=\pm12$. – Lucian Dec 11 '13 at 14:01
This is more memorization of a definition than an actual issue. However the comments/answers here explain several nuances. – Nate 8 Mar 20 at 23:25

While -12 is "a square root" of 144, the square root operation here denotes a function from nonnegative real numbers to nonnegative real numbers. A function can only produce a single result. So the expresion $\sqrt{144}$ evaluates to the positive root 12.

We refer to this convention by saying $\sqrt{x}$ is the principal square root of $x \ge 0$. See the Wikipedia article for more background.

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$144$ has two roots, yes: $\pm 12$. So $-12$ is a root of $144$.

However, we define the principal square root of $\sqrt {x^2} = |x|$, so in your case, $$\sqrt{144} = \sqrt{12^2} = |12| = 12$$

And your calculator is designed to return the principal square root. See Wolfram Alpha for the distinction: $12$ is the principal square root (what we mean by $\sqrt x$, given $x\geq 0$), and $-12$ is considered a real root.

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Needs another TU +1 – Amzoti Dec 12 '13 at 4:37