Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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

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.

share|cite|improve this answer

$144$ 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.

share|cite|improve this answer
Needs another TU +1 – Amzoti Dec 12 '13 at 4:37

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.