Perfect Square Calculator (is it a perfect square? square root & nearest squares)

Understanding the perfect square calculator — concept and principle

This calculator checks whether a whole number is a perfect square, gives its square root, simplifies the root if it is not whole, and shows the nearest perfect squares and a last-digit test. Use it for maths homework, simplifying radicals, factoring, or checking whether a square area has whole-number sides.

A perfect square can be laid out as a square grid; 784 tiles make 28 rows of 28.

How it is calculated

n is a perfect square if n = k² for some whole number k.
The calculator finds the whole-number part of √n exactly and checks whether its square equals n.
If not, it factors n and takes out the largest square factor: √n = k·√m.
It shows the largest square ≤ n, the next square above, and the distance to the nearer one.
Last-digit filter: squares end only in 0, 1, 4, 5, 6 or 9.

Negative numbers are rejected because they have no real square root.

The principle

Squaring a number doubles every exponent in its prime factorisation, so n is a perfect square exactly when every prime exponent is even. 784 = 2⁴ × 7² = (2² × 7)² = 28². The last-digit rule comes from the last digits of 0² to 9²: 0, 1, 4, 9, 6, 5, 6, 9, 4, 1. A number ending in 2, 3, 7 or 8 can never be a square, but passing the test does not prove anything. Consecutive squares differ by 2k + 1, the odd numbers.

Worked example

Default input: n = 784.

√784 = 28, and 28 × 28 = 784 exactly.
Result: Yes, a perfect square: 784 = 28².
784 = 2⁴ × 7², all exponents even.
The next square is 29² = 841, which is 2(28) + 1 = 57 more.
Distance to the nearest square = 0; last digit 4 can be a square, and the root decides.

How to read the result

"Yes" means √n is a whole number, so it is rational. "No" means √n is irrational, and the simplified radical (for example √72 = 6√2) is the exact form. The nearest squares give a quick estimate: a number between 784 and 841 has a square root between 28 and 29.

Common mistakes

Trusting a calculator decimal like 27.99999999; check by squaring a whole number.
Assuming a number ending in 4, 6 or 9 is a square; 14 ends in 4 but is not.
Forgetting that 0 and 1 are perfect squares.
Taking out a square factor that is not the largest, such as √72 = 2√18.

Keep in mind: Inputs must be whole numbers from 0 to 10¹². Fractions such as 9/16 are squares of rationals but are not tested here.

Frequently asked questions

What are the perfect squares up to 100?

0, 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.

Is the square root of a non-square number irrational?

Yes. The square root of a whole number is either a whole number or irrational.

Can a perfect square end in an odd number of zeros?

No. Its zeros come in pairs, so 1,000 cannot be a square.

Sources: OpenStax, Prealgebra 2e (2020), §7.1 Rational and Irrational Numbers · OpenStax, Intermediate Algebra 2e (2020), §8.5

Checked 9 Nov 2026. OpenStax, Prealgebra 2e (2020), §7.1 Rational and Irrational Numbers (square roots of perfect squares are whole numbers; square roots of other whole numbers are irrational). OpenStax, Intermediate Algebra 2e (2020), §8.5 (simplify the radical first, taking out perfect powers of the index).

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