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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

Examples:

Question: What is the square of 7?
What do I do?      Multiply 7 by itself
The math: 72 = 7*7 = 49
The answer: 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
The answer: 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
The answer: 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".

√81

9x9 = 81, so the answer is 9.

{9}
√-16 =

Because you cannot find the square root of a negative number, there is no real square root. So, the answer is simply "no answer."

{no answer}
-√225 =

-15

{-15}
121/64 = (write your answer as a fraction using the division bar, / )

11/8

{11/8}

 32 + 42  =

 32 + 42  =  9 + 16  = √25 = 5

{5}