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.
|√4 OK!||√-4 NOT OK!|
Here are some common square roots.
|12 = 1||√1 = 1|
|22 = 4||√4 = 2|
|32 = 9||√9 = 3|
|42 = 16||√16 = 4|
|52 = 25||√25 = 5|
|62 = 36||√36 = 6|
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.
|The question||What the question is asking you to do|
|What is the square of ___?||Multiply ____ by itself|
|What is the square root of ___?||Find a number that equals ___ when multiplied by itself|
|Question:||What is the square of 7?|
|What do I do?||Multiply 7 by itself|
|The math:||72 = 7*7 = 49|
|Question:||What is the square root of 25?|
|What do I do?||Find a number that equals 25 when multiplied by itself.|
|The math:||25 = WHAT times WHAT
25 = 5 times 5
|Question:||√16 = ?|
|What do I do?||Find a number that equals 16 when multiplied by itself.|
|The math:||16 = WHAT times WHAT
16 = 4 times 4
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”.
9×9 = 81, so the answer is 9.
Because you cannot find the square root of a negative number, there is no real square root. So, the answer is simply “no answer.”
√ 32 + 42
√ 32 + 42
= √ 9 + 16
= √25 = 5