 # How Square Roots Work

Here you’ll be learning about square roots, and I will introduce them to you by using the square. A square is a shape with four equal sides and four corners that are all $90$°. The area of a square is $A=s\cdot s$. If you multiply two equal numbers by each other, you get what is called their square.

When you calculate a square root, you square in reverse. The square root undoes the square. Squares and square roots are opposite operations, a bit like plus and minus. If you want to find the square root of a number $t$, you look for the number $a$ that when multiplied by itself becomes $t$. The symbol for the square root looks like this: $\sqrt{\phantom{t}}$

Rule

### TheConnectionBetweenSquareRootsandSquares

 $\sqrt{t}=\sqrt{{a}^{2}}=\phantom{\rule{-0.17em}{0ex}}{\left(\sqrt{a}\right)}^{2}=a$

Example 1

Square $2$. Set $2$ to the power of $2$, meaning $2$ multiplied by itself:

 ${2}^{2}=2\cdot 2=4.$

Find the square root of $4$ by going in reverse:

 $\sqrt{4}=\sqrt{2\cdot 2}=\sqrt{{2}^{2}}=2$

When you solve the equation ${x}^{2}=4$ you will end up with two solutions, both $2$ and $-2$. In mathematics we write this as $x=±2$. The reason why $-2$ also is a solution is this:

 $\left(-2\right)\cdot \left(-2\right)={\left(-2\right)}^{2}=4$

When you calculate square roots, you have to remember that there are always two solutions. But be aware! If the problem asks you to find the length of a side, you won’t need more than the positive solution. Read the problem carefully - know exactly what it asks of you. It doesn’t make any sense for sides to have negative lengths! We will now take a closer look at square roots and two rules that are useful for both square roots and fractions.

It can be convenient to know the square numbers when you calculate square roots. We want to know these because the square root of a square number gives us an integer. You can get the square number back by multiplying that integer by itself. The name "square numbers" comes from the fact that they are the areas of squares whose lengths are integers.

Rule

### SquareNumbersuptoandIncluding400$\left(s\cdot s={s}^{2}=A\right)$

$\begin{array}{llllllll}\hfill {1}^{2}& =1\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{1}^{2}& =121\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {2}^{2}& =4\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{2}^{2}& =144\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {3}^{2}& =9\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{3}^{2}& =169\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {4}^{2}& =16\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{4}^{2}& =196\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {5}^{2}& =25\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{5}^{2}& =225\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {6}^{2}& =36\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{6}^{2}& =256\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {7}^{2}& =49\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{7}^{2}& =289\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {8}^{2}& =64\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{8}^{2}& =324\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {9}^{2}& =81\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{9}^{2}& =361\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 1{0}^{2}& =100\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}2{0}^{2}& =400\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llllllllllllllll}\hfill {1}^{2}& =1\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}{6}^{2}& =36\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{1}^{2}& =121\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{6}^{2}& =256\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {2}^{2}& =4\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}{7}^{2}& =49\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{2}^{2}& =144\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{7}^{2}& =289\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {3}^{2}& =9\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}{8}^{2}& =64\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{3}^{2}& =169\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{8}^{2}& =324\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {4}^{2}& =16\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}{9}^{2}& =81\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{4}^{2}& =196\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{9}^{2}& =361\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {5}^{2}& =25\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{0}^{2}& =100\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}1{5}^{2}& =225\phantom{\rule{2em}{0ex}}& \hfill \phantom{\rule{1em}{0ex}}2{0}^{2}& =400\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

When you have memorized the square numbers, you will find calculations without a calculator to be much easier. Confidence in mathematics comes from a good understanding of numbers. Learn these square numbers, and you’ll gain a bit of extra time on your tests.

Example 2

What is the square root of $144$?

 $\sqrt{144}=\sqrt{12\cdot 12}=\sqrt{1{2}^{2}}=12,$

because you now know that $12\cdot 12=144$.