# Square Roots

## How do we do them?

A square root of a number
$b$
is a solution of the equation
${x}^{2}=b$
. Every number except
$0$
has two square roots, a positive and a negative. The positive square root is the
**
principal square root
**
and is written
$\sqrt{b}$
. To denote the negative root, write
$-\sqrt{b}$
and to indicate both roots write
$\pm \sqrt{b}$
.

So, we call $5$ the “square root” of $25$ and write $\sqrt{25}=5$ because ${5}^{2}=25$ . (See exponents for more on this.) Since ${\left(-5\right)}^{2}$ also equals $25$ it is also a “square root” of $25$ , but we write $-\sqrt{25}=-5$ because it is not the principal square root.

Not all whole numbers have a whole number square root. For instance $\sqrt{2}=1.414213562\dots $ (The decimal goes on forever and never repeats a pattern. This is called an irrational number .)

How can you estimate the value of a square root like $\sqrt{70}$ ? Well, you could first notice that $\sqrt{64}=8$ and $\sqrt{81}=9$ , since $64$ and $81$ are both perfect squares. $70$ is in-between $64$ and $81$ , so $8<\sqrt{70}<9$ . Since $70$ is closer to $64$ than to $81$ , $\sqrt{70}$ is closer to $8$ .

To find a better approximation, you can use a calculator:

$\sqrt{70}\approx 8.3666$