What the Square Root Calculator does
This square root calculator gives the square root of any whole number or decimal to the number of decimal places you choose, simplifies the radical into the form a√b - so √72 becomes 6√2 - tells you whether the number is a perfect square and which perfect squares sit either side of it, and works the digit-by-digit long-division method step by step, the way it is taught in school.
It is built for square roots as a topic rather than as a button: every step is exact, using whole-number arithmetic on as many digits as you type, so √2 to 50 decimal places is correct to the last digit. For cube roots, fourth roots and other roots as decimals, the separate Nth Root Calculator is the better tool. Everything runs in your browser.
How to use it
- Type the number. Whole numbers and decimals up to 60 digits are accepted; commas between thousands are ignored.
- Choose how many decimal places you want. The last digit is rounded from the next one, not cut off.
- Read the root at the top, then the simplified radical, the perfect-square check and the nearest perfect squares in the tiles.
- The numbered steps show the prime factorisation and which factors come out of the root. Open the long-division table to follow the pairing method one digit at a time.
- Copy the root or the simplified form, or download the long-division steps as CSV for homework notes.
Reading the results
If the simplified radical is a whole number or a terminating decimal, the number is a perfect square and the root is exact. Otherwise the root is irrational: its decimal digits never end or repeat, and the simplified radical (such as 6√2) is the only exact way to write it.
The nearest perfect squares give a quick estimate: 50 lies between 49 and 64, so √50 is a little more than 7. That is a useful check on any answer.
In the long-division table each row produces one digit of the root. The remainder never exceeds twice the root so far, which is a sign that each digit was chosen correctly.
Worked example: simplifying √72 and finding √529 by long division
72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3². One pair of 2s and one pair of 3s come out of the root as 2 × 3 = 6; a single 2 stays inside. So √72 = 6√2 ≈ 6 × 1.414214 = 8.485281. 72 is not a perfect square: it lies between 64 (8²) and 81 (9²).
For √529, pair the digits from the right: 5 | 29. The largest square not above 5 is 4 = 2², so the first digit is 2 and the remainder is 1. Bring down 29 to make 129. Double the root so far (2 → 4) and find d so that 4d × d fits into 129: 43 × 3 = 129 exactly. The root is 23 with remainder 0, so 529 is a perfect square.
√12.25: as a fraction 1225/100, and √1225 = 35, so √12.25 = 35/10 = 3.5.
Formulas and scoring rules
- Simplifying a radical
√(a² × b) = a√bFrom the prime factorisation: each pair of equal primes leaves the root as one factor. Exact for integers up to 10^18 (trial division to the cube root, then an exact square test of the cofactor).- Decimals
√(n / 10^(2m)) = √n / 10^mAn odd number of decimal places is first rewritten with one more, e.g. 0.5 = 50/100, so √0.5 = √50 / 10 = √2 / 2.- Perfect square test
n is a perfect square when floor(√n)² = nfloor(√n) is computed exactly with Newton's method on whole numbers.- Long division (pairing) method
for each pair: c = 100r + pair; choose the largest d with (20p + d) × d ≤ c; r = c − (20p + d)d; p = 10p + dp is the root so far and r the remainder. Digits are exact; the last shown is rounded half up from the next digit.- Negative numbers
√(−x) = i√x
How this differs from the Nth Root Calculator
The Nth Root Calculator finds any root - cube roots, fifth roots, fractional powers - as a decimal. This page is narrower and deeper for square roots: it simplifies radicals exactly, says whether a number is a perfect square, lists the perfect squares on either side, and shows the long-division method that many maths courses still ask students to work by hand.
Why the long-division method works
If the root so far is p and the next digit is d, the new root is 10p + d and its square is 100p² + (20p + d)d. The 100p² part has already been subtracted in earlier steps (shifted two places), so what has to fit into the current remainder is exactly (20p + d) × d. Choosing the largest d that fits produces the root one digit at a time, and taking digits in pairs matches the fact that squaring doubles the number of digits.
Limitations: what the result does not prove
- Radicals are simplified for whole numbers up to 10^18 (and decimals whose digits stay within that). Larger inputs still get an exact root to the chosen decimal places, but no factorisation.
- Only square roots. Use the Nth Root Calculator for cube roots and higher roots.
- Imaginary roots are shown for negative inputs as a multiple of i; complex arithmetic beyond that is not covered.
- Inputs in E notation (such as 2e10) are not accepted, so that every digit you mean is written out.
Privacy: where your data goes
Everything you paste, type or drop is processed in this browser tab. It is not uploaded, logged, stored or sent to analytics. Session recording and tag-manager scripts are switched off on this page.
Standards and sources
- OpenStax Elementary Algebra 2e - Simplify and use square roots
- OpenStax Elementary Algebra 2e - Simplify square roots
- NIST Digital Library of Mathematical Functions - Elementary functions (powers and roots)
Frequently asked questions
How do you simplify a square root?
Factor the number into primes, take one factor out of the root for every pair of equal primes, and leave the unpaired factors inside. √72 = √(2³ × 3²) = 2 × 3 × √2 = 6√2. The calculator shows the factorisation and the pairs.
How do I know if a number is a perfect square?
Its square root is a whole number. The calculator checks this exactly, even for very long numbers, and when the answer is no it gives the perfect squares just below and just above, such as 49 and 64 for 50.
How do you find a square root by long division?
Group the digits in pairs from the decimal point, find the largest square below the first pair, then repeatedly bring down the next pair and find the digit d for which (20 × root so far + d) × d fits. The table shows each of these steps for your number.
What is the square root of 2 to 10 decimal places?
1.4142135624 (rounded; the digits continue 1.41421356237...). Choose more decimal places to see up to 50 digits, all calculated exactly by the digit-by-digit method.
Can you take the square root of a negative number?
Not as a real number, because any real number squared is zero or positive. Mathematicians use the imaginary unit i, where i² = −1, so √(−16) = 4i. The calculator shows that form.
How do you estimate a square root without a calculator?
Find the perfect squares either side. √50 lies between 7 (√49) and 8 (√64), much closer to 7, so about 7.07. For more digits the long-division method gives one exact digit per step.
Last reviewed by the A2Z.Tools team against the sources listed above.
