# Perfect Squares

In mathematics, you will come across different kinds of numbers such as even and odd, prime and composite, and triangle numbers, among many other ways that we group and classify numbers. One type of number you will also run across is a perfect square. A perfect square is a number that can is the result of squaring an integer. Let's look at the definition of a perfect square, the proper notation, and a list of these numbers up to 100.

## Perfect square definition

An integer that can be expressed as the square of another integer n, that is ${n}^{2}$ , is called a perfect square. Since a negative times a negative is a positive, a perfect square is always a positive. To put it another way, a perfect square is defined as the product of any integer with itself.

**Example 1**

${2}^{2}$ , or $2\times 2=4$

Therefore, 4 is a perfect square.

## Perfect square numbers from 1 to 100

Let's take a look at the first 10 perfect squares.

Perfect square numbers from 1 to 100 | ||||

1 | = | $1\times 1$ | = | ${1}^{2}$ |

4 | = | $2\times 2$ | = | ${2}^{2}$ |

9 | = | $3\times 3$ | = | ${3}^{2}$ |

16 | = | $4\times 4$ | = | ${4}^{2}$ |

25 | = | $5\times 5$ | = | ${5}^{2}$ |

36 | = | $6\times 6$ | = | ${6}^{2}$ |

49 | = | $7\times 7$ | = | ${7}^{2}$ |

64 | = | $8\times 8$ | = | ${8}^{2}$ |

81 | = | $9\times 9$ | = | ${9}^{2}$ |

100 | = | $10\times 10$ | = | ${10}^{2}$ |

## Roots of perfect squares

Let's say you were presented with a number, say 67, and you wanted to know if it was a perfect square, you could take out your handy and infinite list of all perfect squares. However, that is time-consuming and tedious, and a better method is to simply take the square root and see if it is an integer. If it is, your number must, by definition, be a perfect square.

$\sqrt{67}$ equals about 8.18

Since 8.18 is not an integer, we know that 67 is not a perfect square. One interesting thing we do learn is that since 8.18 is closer to 8 than 9, the closest perfect square to 67 is ${8}^{2}$ or 64.

## Practice questions on perfect squares

a. Are there any odd perfect squares?

Yes, the square of any odd number is also odd

b. Is 55 a perfect square?

No, the square root of 55 is 7.41

c. Is 144 a perfect square?

Yes, $12\times 12=144$

d. What is the closest perfect square to 19?

$\sqrt{19}$ is about 4.35, which is closer to 4 than 5, so 19 is closer to 16 than it is to 25.

## Topics related to the Perfect Squares

## Flashcards covering the Perfect Squares

## Practice tests covering the Perfect Squares

College Algebra Diagnostic Tests

## Get help learning about perfect squares

If your student is having a difficult time learning how to work with perfect squares, it is a good idea to have them work with a private tutor. A professional tutor can spend 1-on-1 time with your student focusing on the areas of perfect squares that are most challenging to them. They can work through problems step-by-step until your student understands perfect squares thoroughly. To learn more about how tutoring can help your student with perfect squares and other algebra concepts, contact the Educational Directors at Varsity Tutors today.

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