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:

$\sqrt{70}\approx 8.3666$