Perfect Squares — Definition & Examples

(Video) What Is a Perfect Square?

by Brian McLogan

This video by Brian McLogan covers the definition and calculation of perfect squares.

$\large{\text{Summary:}}$

Let's start by looking at a square.

A square has equal sides. If you know one side of the square has a length of $7$, then you know all sides of the square have a length of $7$.

Now, what if we take the area of this square? Remember:

The area of a square is its length times its width.

So, we plug in $l$ and $w$:

$$\begin{align} A&=l\times w\\ A&=7\times7\\ A&=49 \end{align}$$

We say that $49$ is a perfect square because it is the product of an integer multiplied by itself, $7\times7$.

If we had a square with an area of $50$, this wouldn't be a perfect square because no integer, multiplied by itself, will give a product of $50$. In fact, we'd have to resort to using irrational numbers.

To find other perfect squares, we just multiply integers. Positive ones, at that.

$$\begin{align} 2\times2&=4\\ 3\times3&=9\\ 4\times4&=16\\ 5\times5&=25\\ 6\times6&=36\\ 7\times7&=49\\ 8\times8&=64\\ 9\times9&=81\\ 10\times10&=100 \end{align}$$

We can keep going forever and ever. There is an infinite number of perfect squares. If we try to list them all we'd take forever.

A perfect square is any number that can be expressed as the product of two equal integers.

In other words, a perfect square is a number that can be made by squaring a whole number. $\\$ For example, $9$ is a perfect square because $3^2= 3 \times 3= 9$ $\\$ $36$ is also a perfect square because $6^2=6 \times 6= 36$. $\\$ However, numbers like $7$, $15$ and $24$ are not perfect squares because you cannot express them as products of two equal integers. Here are some other perfect squares. $$\begin{align} 1\times1&=1\\ 2\times2&=4\\ 3\times3&=9\\ 4\times4&=16\\ 5\times5&=25\\ 6\times6&=36\\ 7\times7&=49\\ 8\times8&=64\\ 9\times9&=81\\ 10\times10&=100 \end{align}$$

It is worth memorizing at least the first ten.

Let’s try some examples.

Remember $81$ is a perfect square because it can be expressed as the product of two equal integers. $$81= 9 \times 9 = 9^2$$

$45$ is not a perfect square because it cannot be expressed as a product of two equal integers.

Perfect squares and square roots are some of the best introductory material to multiplication.

Let's take a set of coins. Now, let's say you want to arrange them into squares.

Once you play around with the coins for a while, you notice that you can only arrange squares with $4, 9, 16, 25,$ etc. coins. These transfer over to being perfect squares in the mathematical world as well.

For more advanced learners, try to make rectangles or squares out of any number of coins. Multiplication tells you which rectangles you can make (i.e. a rectangle made with $3$ rows of $5$ coins can be made with $15$ coins. Any number that you can't arrange into a rectangle with at least $2$ rows and $2$ columns is a prime number.

Perfect Squares

Let's think about what the definition of a "perfect square" is. When you are drawing a perfect square, what property makes it "perfect"? Well, it is perfect when each of the $4$ sides have the same length!

Now, the context in which we are talking about perfect squares is not the same as physically drawing them. Here, we are talking about performing the inverse of an exponent. So, when you raise a number to an exponent, $n$, you multiply that number by itself $n$ times. When $n = 2$, we call that squaring a number. Just like drawing a perfect square, where all the sides are the same length, when you square a number, you are taking the same number and multiplying it with itself! Therefore, the perfect squares are the numbers that are the resultant of squaring a number! Let's list some of them below:

$$\begin{align} 1^2 &= 1 \\ 2^2 &= 4 \\ 3^2 &= 9 \\ 4^2 &= 16 \\ 5^2 &= 25 \\ 6^2 &= 36 \\ 7^2 &= 49 \\ 8^2 &= 64 \\ 9^2 &= 81 \\ 10^2 &= 100 \\ \end{align}$$

Now, it is important to note that the perfect squares are the numbers on the right of the equal sign. So, our perfect squares are $1, 4, 9, 16, 25, 36, 49, 64, 81,$ and $100$.