Perfect Square Calculator

What is a Perfect Square?

A perfect square (also known as a square number) is an integer that can be expressed as the product of an integer with itself. In mathematical notation, if \(n\) and \(k\) are integers such that:

$$n = k^2 = k \times k$$

then \(n\) is a perfect square, and \(k\) is the square root of \(n\), written as \(\sqrt{n} = k\).

Because squaring any real integer (positive, zero, or negative) always results in a non-negative value, all real perfect squares are greater than or equal to 0 (\(n \ge 0\)).

Table of the First 25 Positive Perfect Squares

The table below lists the squares for base numbers 1 through 25:

Properties and Quick Tricks to Check Perfect Squares

1. Last Digit Rule (Units Digit)

In base 10, a perfect square can only end in 0, 1, 4, 5, 6, or 9. If an integer ends in 2, 3, 7, or 8, it is guaranteed not to be a perfect square.

2. Digital Root Rule (Mod 9)

The digital root (repeated sum of digits until a single digit remains) of any perfect square must be 1, 4, 7, or 9 (or 0 for zero). For example, \(144 \rightarrow 1 + 4 + 4 = 9\).

3. Prime Factorization Exponents

When factored into prime numbers, a number is a perfect square if and only if every prime factor exponent is an even integer. For instance, \(144 = 2^4 \times 3^2\). Both exponents (4 and 2) are even, so 144 is a square. You can also inspect cubic numbers with our Perfect Cube Calculator or compute square roots directly using our Square Root Calculator.