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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 theprincipal square rootand 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 anirrational 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:

$\sqrt{70}\approx 8.3666$