# What is the square root of (-12)^2?

Apr 11, 2018

The square root of anything squared is itself, almost always.

#### Explanation:

When you square something, essentially you are multiplying it by itself. For instance, ${2}^{2} = 2 \cdot 2 = 4$, and $\sqrt[2]{4} = 2$, therefore . In your scenario, we’re doing $\left(- 12\right) \cdot \left(- 12\right)$. However, as you’ve probably learned, a negative times a negative is a positive! What now? There are a few ways we could go with this:

Way one: we assume that every square root will be positive. This is the easiest way, but it’s not the most accurate. In this case, the answer to $\sqrt[2]{- {12}^{2}}$ would be $12$, because $\left(- 12\right) \cdot \left(- 12\right) = 144$, and $\sqrt[2]{144} = 12$.

Way two is only a little bit more complicated. We assume that every square root could be either negative or positive, so the answer to $\sqrt[2]{- {12}^{2}}$ would be $\pm 12$, because $\left(- 12\right) \cdot \left(- 12\right) = 144$ and $12 \cdot 12 = 144$, so $\sqrt[2]{144}$ could equal either $+ 12$ or $- 12$, and the way that is written in math notation is $\pm 12$.

Apr 11, 2018

#### Explanation:

The question makes an assumption that is not, in general, warranted.

The phrase "the square root" indicates that only one answer is expected.

Now we might assume that the real question is "What is the principal square root of ${\left(- 12\right)}^{2}$?" In this case, since the principal square root or a positive number is the non-negative square root, the answer is $12$.
Note that for non-negative real $n$, the symbol $\sqrt{n}$ always refers to the principal square root.

The definition of a square root is:

$a$ is a square root of $b$ if and only if ${a}^{2} = b$.

So every positive number has 2 square roots. It has a positive square root (the principal square root) and a negative square root.

The two square roots of ${\left(- 12\right)}^{2}$ are $12$ and $- 12$

$12$ is a square root of $144$ and $- 12$ is a square root of $144$

The two solutions two ${x}^{2} = {\left(- 12\right)}^{2}$ are the square roots of $144$. They are $\sqrt{144}$ and $- \sqrt{144}$