# Introduction to Square Roots

### Written by tutor April G.

A square root is a number a such that for a number b, b^{2} = 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, 3^{2} = 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.

1^{2} = 1 |
√1 = 1 |

2^{2} = 4 |
√4 = 2 |

3^{2} = 9 |
√9 = 3 |

4^{2} = 16 |
√16 = 4 |

5^{2} = 25 |
√25 = 5 |

6^{2} = 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: | 7^{2} = 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”.

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.”

-15

^{121}/

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

^{11}/_{8}

√ 3^{2} + 4^{2}

=

√ 3^{2} + 4^{2}

= √ 9 + 16

= √25 = 5