What it does
The square root of a number is what you multiply by itself to get it: √49 = 7 because 7 × 7 = 49. Picture a square with that area: the square root is the length of its side. This page finds square roots four ways and shows every step, so you can follow the method rather than just press a button.
- Prime factors: split the number into primes and pair up equal ones. One prime from each pair multiplies to the root. If a prime is left without a partner, the number is not a perfect square, and the page writes the root in its simplest surd form, like √72 = 6√2.
- Odd numbers: take away 1, 3, 5, 7, … in turn. Landing exactly on 0 means a perfect square, and the number of steps is the root (because the first n odd numbers add up to n²).
- Estimate: find the two square numbers either side, share out the gap for a first guess, then improve it with new guess = (guess + number ÷ guess) ÷ 2. Each step roughly doubles the number of correct digits.
- Long division: the pencil-and-paper method that gives one digit at a time, and also works for decimals.
It follows Class 8 Maths, a square and a cube (square roots by prime factorisation, repeated subtraction and estimation), and helps with surds in Class 9 (simplifying √72 = 6√2). For pictures of squares and cubes growing, open the squares and cubes; for prime factor trees, HCF and LCM; and for where square roots turn up in right triangles, the pythagoras proof playground playground.
How to use it
- Type a number in the box (whole or decimal, like 7056 or 12.25), or tap a quick example.
- The picture shows a square of that area and a number line with the two nearest square numbers. The big line gives the root, or says it is not a perfect square.
- Pick a method: Prime factors, Odd numbers, Estimate or Long division. The steps update as you type.
Worked example
The page opens with 7056.
- Prime factors: 7056 = (2 × 2) × (2 × 2) × (3 × 3) × (7 × 7). Every prime is in a pair, so √7056 = 2 × 2 × 3 × 7 = 84.
- Odd numbers: 7056 − 1 = 7055, 7055 − 3 = 7052, … after 84 subtractions it reaches exactly 0.
- Long division: pairs 70 | 56. The biggest digit whose square fits in 70 is 8 (64), leaving 6. Bring down 56 to get 656; the divisor is 2 × 8 = 16, and 164 × 4 = 656 exactly. So √7056 = 84.
Now try 40: it lies between 36 = 6² and 49 = 7², so the first guess is 6 + 4 ÷ 13 = 6.3077. One improving step gives 6.324573 and the next 6.324555, already right to six decimal places. And 72 = (2 × 2) × (3 × 3) × 2: the last 2 has no partner, so √72 = 2 × 3 × √2 = 6√2 ≈ 8.4853.
Check yourself
Five fresh questions: the root of a perfect square, the number of odd-number steps, a simplest surd form, the two whole numbers a root lies between, and the root of a decimal. Nothing is saved.
FAQ
Why can a square root not end in a remainder? For a perfect square, every method ends exactly. If the long division keeps leaving a remainder, the root is an irrational number whose decimals never end or repeat, like √2 = 1.41421356…
Which method should I use in an exam? Prime factors for perfect squares you can factor; long division for big numbers and decimals; estimation to check that an answer is sensible.
Is this from the NCERT book? It covers the same topics with fibub’s own pictures and examples. fibub is not part of NCERT.
Where the numbers come from
- Square roots by prime factorisation, by repeated subtraction of odd numbers, by estimation between squares, by Heron's (Babylonian) method x → (x + n/x)/2 and by the long division method are standard school arithmetic. Wording, pictures and examples are fibub's own.
- Long division digits checked with exact integer arithmetic against Python's math.isqrt for every whole number to 100,000 and for decimals such as 2, 10 and 12.25.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.