Introduction to Square Roots

Written by tutor April G.

A square root is a number a such that for a number b, b² = a. In other words, a number b whose square is a.
Another way to say this is that a square root of a number is one of its two equal factors.

For example, 3² = 3*3 = 9

3 is considered a square root of 9, because when 3 is multiplied by itself it equals 9. We write this as √9 = 3
-3 is ALSO considered a square root of 9 because (-3)*(-3) = 9. Therefore, √9 = -3 as well.

Because of this, every non-negative real number has 2 square roots, a positive (or principal) square root and a negative square root.
To eliminate confusion, generally we write a negative sign to specify the negative root and either a positive sign (or no sign) when
talking about the positive root. If we want to talk about both roots of a number a, we would write ±√a.

(By the way, that √ symbol is called a radical. You’ll learn in later math more about radicals, but for now we’re just going
to talk about square roots.)

You’ll note I said non-negative real number. There’s no such real number, for example, of the square root of -9 (√-9).
It makes sense if you think about it: you always get a positive number when you multiply two numbers with the same sign.
3*3 = 9 and (-3) * (-3) = 9, but neither gives you a negative 9!

In fact, we call the square root of a negative number an imaginary number, which is used when talking about complex numbers.
But that’s a topic for another day. For now, just remember that you can’t take the square root of a negative number.

Here are some common square roots.

Numbers like 1, 4, 9, 16, etc, are called perfect squares because they are the squares of integers. The numbers in-between,
like 15 or 27, are not perfect squares. Square roots of these numbers are called irrational numbers. If you use a calculator to find √15,
for example, will give you 3.872983346207417…. You can also find square roots of fractions. For example, because ²/₃ x
²/₃ = ⁴/₉, ²/₃ is a square root of ⁴/₉. An easy way of
looking at this is looking at the square roots of the numerator and denominator separately.

Example: Find √²⁵/₃₆. Solution: Because the square root of 25 is 5, and teh square root of 36 is 6, then the square root of
²⁵/₃₆ = ⁵/₆.

It’s important to know the difference between the questions “What is the square of ____?” and “What is the square root of ____?” In one case you are
taking the number and multiplying it by itself, and in the other you are finding the number’s square root.

Examples:

Finally, when evaluating expressions with square roots, treat the radical the same way you would treat parenthesis. So to evaluate
√ 5 + 4 
-1, first you would add under the radical (5+4=9), then evaluate the radical (√9 = 3), and finally subtract 1 to get 2.

Square Roots Practice Quiz

Here are some additional examples to try. If there is no possible answer, simply type “no answer”.