Perfect Square Calculator

About Perfect Square Calculator

A perfect square is an integer that is the square of another integer. 1, 4, 9, 16, 25, ... are perfect squares.

How to use this calculator

1. Enter the number you want to test, such as 64 or 50.
2. Check the result to see whether the number is a perfect square.
3. If it is, read the integer square root shown by the calculator.
4. Use the explanation to understand why the number does or does not match a square like 8^2 or 9^2.

The formula explained

The formula says that a number \(n\) is a perfect square exactly when it can be written as \(k^2\) for some integer \(k\). In that case, \(\sqrt{n}\) is also an integer.

• n = the number being tested
• k = an integer whose square equals n
• \(\sqrt{n}\) = the square root of n

Step by step method

1. Start with the number you want to check.
2. Find an integer whose square might match it, or take the square root and see whether it is a whole number.
3. If the square root is an integer, the number is a perfect square, and that integer is the root. If not, it is not a perfect square.

Worked example

Problem. Check whether 81 is a perfect square and find its root.

1. Test nearby squares: 8^2 = 64 and 9^2 = 81.
2. Since 81 = 9^2, the square root is \(\sqrt{81} = 9\).
3. Because the root is an integer, 81 is a perfect square.

Tips and common mistakes

• Only whole numbers count as perfect squares, so 12.25 is not a perfect square even though its square root is 3.5.
• If the square root has a decimal, the number is not a perfect square. For example, \(\sqrt{50}\) is not an integer.

Frequently asked questions