What Is a Perfect Square



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What is a perfect square? How do you know if a number is a perfect square?

In this article, we’ll define perfect squares, provide a list of the first 25 perfect squares (and the integers that make them up), and teach you how to tell if a number is a perfect square.

What Is a Perfect Square?



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

What does that mean?

Basically, a perfect square is what you get when you multiply two equal integers by each other.

For instance:

$$5 * 5 = 25$$

25 is a perfect square because you’re multiplying two equal integers (5 and 5) by each other.

You can also express $5 * 5$ as $5^2$. That’s where you get the term “perfect square”

List of Perfect Squares


Here’s a list of the first 25 perfect squares. A hint: if you want to create a perfect square, simply square an integer!


Square

Integers
1 $1 * 1$
4 $2 * 2$
9 $3 * 3$
16 $4 * 4$
25 $5 * 5$
36 $6 * 6$
49 $7 * 7$
64 $8 * 8$
81 $9 * 9$
100 $10 * 10$
121 $11 * 11$
144 $12 * 12$
169 $13 * 13$
196 $14 * 14$
225 $15 * 15$
256 $16 * 16$
289 $17 * 17$
324 $18 * 18$
361 $19 * 19$
400 $20 * 20$
441 $21 * 21$
484 $22 * 22$
529 $23 * 23$
576 $24 * 24$
625 $25 * 25$

How to Tell If a Number Is a Perfect Square


You can tell if a number is a perfect square in a couple of different ways.

First of all, if you create a square by multiplying two equal integers by each other, then the product is a perfect square.

So, $1 * 1$ is a perfect square. So is $10 * 10$ and $1,000 * 1,000$.


You can also tell if a number is a perfect square by finding its square roots.

Finding the square root is the inverse (opposite) of squaring a number.


If you find the square root of a number and it’s a whole integer, that tells you that the number is a perfect square.

For instance, the square root of 25 is 5. The square root of 26 is not a whole integer. So, 26 is not a perfect square.

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Key Takeaways: Understanding Perfect Squares


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

You can tell if a number by finding its square root and seeing if that square root is a whole integer.

What’s Next?


Getting ready to take the ACT?

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A good place to start is mastering systems of equations.