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Perfect Squares: Definition and Examples | EDU.COM

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Perfect Squares: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Perfect SquaresPerfect Squares: Definition and ExamplesTable of ContentsPerfect Squares Definition of Perfect Squares

A perfect square is a positive integer that can be written as the product of an integer multiplied by itself. In other words, when we multiply any integer by itself, we get a perfect square number. We can write perfect squares using the second exponent of an integer, such as n2n^2n2. For example, 161616 is a perfect square because it equals 4×44 \times 44×4 or 424^242. It's also worth noting that the square of both positive and negative numbers can result in the same perfect square (e.g., (−4)×(−4)=16(-4) \times (-4) = 16(−4)×(−4)=16).

Perfect squares can be visualized as numbers that can form a complete square shape when arranged in rows and columns. For example, 999 marbles can be arranged in 333 rows and 333 columns to form a square shape. However, numbers like 666 cannot form perfect squares because they can only be arranged in uneven arrays (like 222 rows and 333 columns). All perfect squares end in 000, 111, 444, 555, 666, or 999, and numbers ending in 222, 333, 777, or 888 cannot be perfect squares.

Examples of Perfect Squares Example 1: Finding the Perfect Square of 131313 Problem:

Find the perfect square of 131313 using the formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2(a+b)2=a2+2ab+b2.

Step-by-step solution:

Step 1, Split 131313 into friendly numbers to make calculation easier. We can write 131313 as (10+3)(10 + 3)(10+3).

Step 2, Apply the square formula to our split numbers. We know that (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2(a+b)2=a2+2ab+b2, so for (10+3)2(10 + 3)^2(10+3)2:

132=(10+3)213^2 = (10 + 3)^2132=(10+3)2 =102+2(10)(3)+32= 10^2 + 2(10)(3) + 3^2=102+2(10)(3)+32

Step 3, Solve each part of the equation.

=100+60+9= 100 + 60 + 9=100+60+9

Step 4, Add up all the parts to get the final answer.

=169= 169=169

So, the perfect square of 131313 is 169169169.

Example 2: Checking if a Number is a Perfect Square Problem:

Determine if 3,6003,6003,600 is a perfect square or not.

Step-by-step solution:

Step 1, Look at the end of the number. Since 3,6003,6003,600 ends with two zeros (an even number of zeros), it might be a perfect square.

Step 2, Try to break down the number into factors.

3,600=6×6×10×103,600 = 6 \times 6 \times 10 \times 103,600=6×6×10×10

Step 3, Rearrange the factors to see if they form a perfect square pattern.

3,600=6×6×10×10=60×60=6023,600 = 6 \times 6 \times 10 \times 10 = 60 \times 60 = 60^23,600=6×6×10×10=60×60=602

Step 4, Since we can write 3,6003,6003,600 as 60260^2602, we confirm that 3,6003,6003,600 is a perfect square.

Example 3: Finding the Number to Add to Make a Perfect Square Problem:

What is the smallest whole number to be added to 555555 to make it a perfect square?

Step-by-step solution:

Step 1, Think about what we're trying to do. We need to find a perfect square that is slightly larger than 555555.

Step 2, Look at nearby perfect squares. We know that 72=497^2 = 4972=49 (which is too small) and 82=648^2 = 6482=64 (which is larger than 555555).

Step 3, Calculate how much needs to be added to 555555 to reach the nearest perfect square above it.

64−55=964 - 55 = 964−55=9

Step 4, Check our answer: 55+9=64=8255 + 9 = 64 = 8^255+9=64=82, which is a perfect square. So, the smallest whole number to add to 555555 to make it a perfect square is 999.

Comments(4)TTherapistVinceNovember 6, 2025This glossary def on perfect squares is great! I've used it to explain to my students, and they finally get it. Thanks!

BBasketballPlayerLunaNovember 5, 2025I've used this perfect squares def. with my students. It's super clear and really helped them grasp the concept. Thanks!

PPoetMonaNovember 4, 2025I've used this perfect squares def to help my students. It's clear & the examples make it easy for them to grasp. Great resource!

CCanoerCindyNovember 4, 2025I've used this perfect squares def to help my students. It's clear & the examples really make the concept easy to grasp!

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