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Jul 17, 2024 | Almaden

This easy-to-follow guide to perfect squares in math is here to help you out, whether you're just learning about them or need a quick reminder. You’ll discover easy explanations, helpful tips, and fun activities!

For a more complete overview of the topic, we recommend visiting the *Part 1: What Does Squared Mean in Math?* guide.

**A perfect square** — or simply, **a square** — is a number that we get when we multiply a whole number by itself.

Let's see how that works in numbers:

When we multiply 3 by itself, which can be expressed with 3×3 or 3^{2}, we get 9.

So, 9 is a perfect square.

We can square any whole number, i.e. multiply it by itself, and get a perfect square:

- 4
^{2 }= 16 - 5
^{2 }= 25 - 6
^{2 }= 36 - And so on

In these examples, the numbers 16, 25, and 36 are all perfect squares.

We can present perfect squares visually by treating each whole number that builds into it as a row of little blocks, like so:

Now let’s count the blocks:

- To square the number 2, we place 2 blocks into 2 rows, and we get 4 blocks in total. 4 is a perfect square.
- To square the number 3, we place 3 blocks into 3 rows, and we get 9 blocks in total. 9 is a perfect square.
- To square the number 4, we place 4 blocks into 4 rows, and we get 16 blocks in total. 16 is a perfect square.
- To square the number 5, we place 5 blocks into 5 rows, and we get 25 blocks in total. 25 is a perfect square.
- And so on.

As you can see, placing the blocks into an equal number of rows forms a perfect square. To get our square number, all we have to do is count the blocks.

Here are some more examples of perfect squares.

The easiest way to determine whether a number is a perfect square or not is by finding its square root. A square root is the number we have to multiply by itself to get a square number.

The square root of a perfect square is always **a whole number**.

Take the number 25.

The square root of 25 is 5 because 5x5 = 25.

Since 5 is a whole number, 25 is a perfect square.

Now, let's try the number 30.

What number do we have to multiply by itself to get 30? In other words: What is the square root of 30?

That number is approximately 5.477 (we read this as "five and four hundred seventy-seven thousandths"). Whenever you see the decimal point in a number, you will know that it is not a whole number.

So, the number 5.477 is **not a whole number.**

As we said, the root of a perfect square is a whole number, so we can conclude that **the number 30 is** **not a perfect square**.

Another useful clue for determining if a number is a perfect square or not is to look at its last digit.

Go back to our list of perfect squares and notice that none of the **numbers end in 2, 3, 7, or 8**.

Let’s test this on large numbers:

- Numbers 40
**2**, 41**2**, 42**2**, 43**2**, etc., are not perfect squares. For instance, the square root of 402 is approximately 20.05 which is not a whole number. - Let's try the numbers ending in 3, such as 40
**3**, 41**3**, 42**3**, 43**3**, etc. For example, the square root of 413 is approximately 20.32, which is not a whole number and, therefore, 413 is not a perfect square. - Let's try the numbers ending in 7, such as 40
**7**, 41**7**, 42**7**, 43**7**, etc. For example, the square root of 407 is approximately 20.17, which is not a whole number and, therefore, 407 is not a perfect square. - Lastly, looking at numbers like 40
**8**, 41**8**, 42**8**, etc., none of these are perfect squares either. The square root of 408 is approximately 20.2 which is not a whole number, so 408 is not a perfect square.

Although not a rule, one more pattern you can find by observing the last digits of **perfect squares** is that they **might end in numbers like 1, 4, 5, 6, 9, and 00.**

Examples of this include:

121 = 11 x 11

324 = 18 x 18

625 = 25 x 25

196 = 14 x 14

169 = 13 x 13

400 = 20 x 20

Check out **our video** for a full guide to finding perfect squares and square roots.

By now, you know that the most common path to finding perfect squares is to multiply a whole number by itself. When we need to square a multi-digit number, we use the long multiplication method.

For instance, we can use the long multiplication method to square 23.

**Step 1:**Write the number you want to square. Then, write it again below.**Step 2:**Start by multiplying the ones digit of the bottom number (3) by the top number (23).**Step 3:**Now, multiply the tens digit of the bottom number (2) by the top number (23), but start one place to the left and add a zero to the right of the result.**Step 4:**Finally, add up both of the partial products to get your final answer.

Pretty helpful, right?

However, this process gets more complicated with larger numbers. That’s why our tutors have put together a couple of tricks to help you find perfect squares more quickly:

Let’s choose a number ending in 5, like 35.

To find the perfect square of 35, we’ll follow these simple steps:

- Step 1: Separate 3 and 5.
- Step 2: Multiply 3 by its successor, number 4.
**3 x 4 = 12** - Step 3: Square number 5.
**5² = 25** - Step 4: Put the digits of the second step (12) and the third step (25) together.
**You get 1225.**

This method might be interesting to high school students who have already mastered the trinomial formula. To refresh our memory, the trinomial formula looks like this:

*(a + b) ^{2} = a^{2} + 2ab + b^{2}*

This method breaks a large number into smaller, more manageable parts, and helps us figure out the answer without having to take too many steps.

Now, let’s illustrate this formula with the same number as in the previous section, 35.

We can express 35 as a sum of two numbers, 30 and 5. So in this case *a = 30*, *b = 5*

*35 ^{2 }= (30 + 5)^{2} = 30^{2} + 2 (30) (5) + 5^{2}*

* = 900 + 300 + 25*

*= 1225*

Perfect squares are not just limited to textbook pages and classroom walls. They are used in countless ways in our daily lives, especially when we are designing, constructing, and organizing spaces and objects.

Minecraft

Minecraft, the wildly popular sandbox game, is renowned for its limitless creative potential.

Within its pixelated landscapes, players can construct anything their imaginations can come up with, from towering castles to intricate redstone contraptions. At the heart of many of these creations lies the perfect square.

**Building a House**: In Minecraft, when you want to make a house, you start by putting down square blocks in a line to make the bottom part.**Making Walls and Rooms**: After you have the bottom part, you add more blocks on top to make walls. Each wall is like a big square, and you can make different rooms by adding walls in different places.**Adding Floors and Ceilings**: When your walls are up, you can make floors and ceilings by adding more square blocks. This helps make each level of the house fit together nicely.

The Rubik's Cube is another example of perfect squares in action. It's made up of smaller squares arranged in a 3x3 grid on each face, challenging players to solve its colorful puzzle by twisting and turning its sides.

Sudoku puzzles are filled with perfect squares! The puzzle grid consists of nine rows and nine columns, forming nine smaller 3x3 squares within it. The goal is to fill each row, column, and 3x3 square with the numbers 1 through 9 without repeating any.

Example 1

Let’s square number 85 using the method our tutors provided.

- Step 1: Separate 8 and 5.
- Step 2: Multiply 8 by its successor, 9.
**8 x 9 = 72** - Step 3: Square number 5.
**5² = 25**

Using the perfect square trinomial formula, let’s find the square of 52. We can express 52 as a sum of two numbers, 50 and 2. So in this case a = 50, b = 2

The perfect square trinomial formula is: *(a + b) ^{2 }= a^{2 }+ 2ab + b*

*52 ^{2 }= (50 + 2)^{2} = 50^{2} + 2(50)(2) + 2^{2}*

*52 = 2500 + 200 + 4*

*52 ^{2} = 2704*

A) 16

B) 25

C) 32

A) Numbers ending in 3, 6, or 9

B) Numbers ending in 2, 7, or 8

C) Numbers ending in 1, 4, or 5

- 13
- 16
- 14

A) 1,4,6

B) 1,4,9

C) 1,6,9

Keep scrolling to find the quiz answers!

Find answers to common questions about the definition, properties, and examples of perfect squares.

Perfect squares are never negative numbers. They're always positive whole numbers because they show the size of a square, and you can't have negative sizes.

When we multiply 0 by itself, we always get 0. Similarly, the square root of 0 is also 0. That's why we think of it as a perfect square.

Yes, but the square roots are not whole numbers. In fact, they are not even terminating or repeating decimals like 0.5, 0.75, or 0.6.

The square root of a non-perfect square is an irrational number.

The decimal representations of irrational numbers never terminate or repeat.

Example:

*√2=1.41421356237309504880… *

Perfect square trinomials aren’t usually part of the middle school math curriculum unless you are an advanced student. Nevertheless, it is a question related to perfect squares, so let’s explore it!

In algebra, our knowledge of perfect squares helps us to simplify expressions and solve equations by recognizing patterns like the trinomial formula we’ve seen earlier, which allows us to factor quadratic expressions and solve quadratic equations more efficiently.

**A perfect square trinomial** is an algebraic expression that comes from squaring a binomial. A binomial is the addition or subtraction of two terms, like this:

*(x + 3).*

To get a perfect square trinomial, you multiply the binomial by itself:

**Step 1:** Multiply the binomial by itself:

*(x + 3)**(x + 3)*

**Step 2**: Multiply each term in the binomial:

*= x * x + x * 3 + 3 * x + 3 * 3*

*= x ^{2 }+ 6x + 9*

The final expression, *x ^{2 }+ 6x + 9*
, is a perfect square trinomial.

There are two common formulas for perfect square trinomials:

*(a + b) ^{2 }= a^{2 }+ 2ab + b*

*(a - b) ^{2} = a^{2} - 2ab + b^{2}*

Mathnasium’s specially trained tutors work with students of all skill levels to help them learn and master any K-12 math topic, including perfect squares.

Our tutors assess each student’s skills to develop personalized learning plans that will put them on the best path towards math excellence.

Find a Mathnasium Learning Center near you, schedule an assessment, and enroll today!

^{1. Not approved by or associated with Mojang or Microsoft.}

^{2. Quiz Answers: 1 - C, 2 - B, 3 - C, 4 - B.}