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How to Find the Square Root of Any Number by the Long Division Method

Some square roots are easy. You see 49 and you just know it’s 7. Then a test paper throws 5476 or 71 at you, and suddenly guessing doesn’t cut it.

That’s when the square root by division method earns its place. It works on any number: huge, tiny, perfect square, not-so-perfect square, even decimals. Is it slower than a calculator? Yes. But after a few rounds of practice it starts to feel almost automatic, because you’re not being clever, you’re just repeating the same little loop until the answer appears.

By the end of this guide you’ll know how the method works and why, how to run it step by step, and how to handle perfect squares, non-perfect squares and decimals. You’ll also know the slip-ups that quietly cost students marks, and how to check your work in about ten seconds.

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How to find square root of any number by long division method
☰ Table of Contents

    What is the square root by division method?

    Quick refresher first. The square root of a number is whatever you multiply by itself to get that number back. Since 12 × 12 = 144, we say √144 = 12.

    The long division method (you’ll also see it called the division method or the digit-by-digit method) builds that answer one digit at a time. Each round you find the biggest digit that still fits, subtract, bring down the next chunk of the number, and go again.

    Why bother with it when other methods exist?

    • Prime factorisation is lovely for perfect squares, but it gets you nowhere with a number like 71.
    • Estimation gets you close, but “close” doesn’t earn full marks.
    • Long division works on everything, and you can keep going to as many decimal places as you like.

    If your textbook is covering squares, radicals or number systems, this is almost certainly the method it expects for bigger numbers.

    A few words you’ll see along the way

    Don’t worry about memorising these. Just skim them so nothing feels foreign later.

    • Radicand: the number sitting under the root sign. In √529, it’s 529.
    • Perfect square: a whole number that’s the square of another whole number (1, 4, 9, 16, 25, 36…).
    • Non-perfect square: a number like 2, 71 or 128. Its square root is a decimal that never ends and never repeats.
    • Pair (or period): a group of two digits. The method chops your number into these.
    • Divisor and quotient: same meaning as in ordinary long division. The quotient is the answer you’re building up.

    The method, step by step

    Here’s the whole recipe. Read it once, then see it in action in the examples, because it makes a lot more sense when there are real numbers involved.

    1. Pair up the digits. Start at the decimal point and work outwards. For a whole number, that means pairing from the right, so if there’s an odd number of digits, the leftmost group will be a lone digit. For decimals, pair the digits after the point from left to right, and add a zero if the last pair is incomplete.
    2. Deal with the first group. Find the biggest number whose square fits into it (is less than or equal to it). That’s the first digit of your answer, and also your first divisor.
    3. Subtract. Multiply, subtract, and you’ve got a remainder.
    4. Bring down the next pair and stick it next to the remainder. That’s your new dividend.
    5. Double the answer so far. Write it down with a blank beside it, like 4_.
    6. Fill in the blank. Pick the biggest digit that makes (your number with that digit) × (that digit) fit into the dividend.
    7. Subtract, then repeat.
    8. Stop when you hit a remainder of zero (a perfect square) or when you’ve got as many decimal places as you need.

    If you want one line to remember: divide, multiply, subtract, bring down, repeat.

    Example 1: A perfect square (529)

    Pair the digits: 529 becomes 5 | 29.

    First group: the biggest square that fits into 5 is 4 (2 × 2). So the first digit of the answer is 2, and 5 − 4 leaves 1.

    Bring down 29 and you’ve got 129.

    Double the answer so far: 2 doubled is 4, so the new divisor is 4_.

    Find the digit: try 3. 43 × 3 = 129, which is exactly what we need. The answer is now 23.

    Subtract: 129 − 129 = 0.

    Nothing left over, so √529 = 23. Done.

    Example 2: A bigger number (7056)

    Pair it up: 70 | 56.

    • The biggest square that fits into 70 is 64 (8 × 8). Answer so far: 8. Remainder: 70 − 64 = 6.
    • Bring down 56 to get 656.
    • Double 8 to get 16, so the divisor is 16_.
    • 164 × 4 = 656. Answer: 84. Remainder: 0.

    So √7056 = 84.

    Notice that a four-digit number needed only two rounds. That’s not a coincidence: the number of pairs always matches the number of digits in your answer.

    Example 3: A non-perfect square (71)

    This is where the method really shows its worth. Since 71 isn’t a perfect square, the remainder won’t ever settle at zero, so we’ll stop at three decimal places.

    Write 71 as 71 . 00 00 00, adding three pairs of zeros for three decimal places.

    • The biggest square that fits into 71 is 64. Answer so far: 8. Remainder: 7.
    • Put the decimal point in the answer. Bring down 00 to get 700.
    • Double 8 to get 16, so the divisor is 16_. Try 4: 164 × 4 = 656. Remainder: 44. Answer: 8.4.
    • Bring down 00 to get 4400. Double 84 to get 168, so the divisor is 168_. Try 2: 1682 × 2 = 3364. Remainder: 1036. Answer: 8.42.
    • Bring down 00 to get 103600. Double 842 to get 1684, so the divisor is 1684_. Try 6: 16846 × 6 = 101076. Remainder: 2524. Answer: 8.426.

    So √71 ≈ 8.426, correct to three decimal places.

    Example 4: A decimal number (12.25)

    With decimals, pair the digits on each side of the point separately: 12 . 25.

    • The biggest square that fits into 12 is 9. Answer so far: 3. Remainder: 3.
    • Put the decimal point in the answer and bring down 25 to get 325.
    • Double 3 to get 6, so the divisor is 6_. Try 5: 65 × 5 = 325. Remainder: 0.

    So √12.25 = 3.5.

    Here’s a neat way to sanity-check decimal answers: the root has half as many decimal places as the original number. 12.25 has two, and 3.5 has one.

    Example 5: The square root of 2

    This one turns up everywhere, so it’s worth getting comfortable with. Write 2 as 2 . 00 00 00.

    • The biggest square that fits into 2 is 1. Answer: 1. Remainder: 1.
    • Bring down 00 to get 100. Double 1 to get 2, so the divisor is 2_. Try 4: 24 × 4 = 96. Remainder: 4. Answer: 1.4.
    • Bring down 00 to get 400. Double 14 to get 28, so the divisor is 28_. Try 1: 281 × 1 = 281. Remainder: 119. Answer: 1.41.
    • Bring down 00 to get 11900. Double 141 to get 282, so the divisor is 282_. Try 4: 2824 × 4 = 11296. Remainder: 604. Answer: 1.414.

    So √2 ≈ 1.414. You could keep going all day and the digits would never fall into a repeating pattern, which is one of the signs that √2 is irrational.

    Example 6: Finding the least number to subtract

    Exam setters love this type of question. What’s the smallest number you must subtract from 1500 to get a perfect square?

    Run the division on 1500, paired as 15 | 00:

    • The biggest square that fits into 15 is 9. Answer so far: 3. Remainder: 6.
    • Bring down 00 to get 600. Double 3 to get 6, so the divisor is 6_. Try 8: 68 × 8 = 544. Remainder: 56. Answer: 38.

    The remainder is 56, which means 38² sits exactly 56 below 1500. Subtract it:

    1500 − 56 = 1444, and √1444 = 38.

    Answer: 56.

    What if the question asked what to add instead? Then you’d head for the next perfect square. 39² = 1521, and 1521 − 1500 = 21.

    A few shortcuts that actually help

    • Know your squares up to about 20. It makes the first step of almost every problem instant.
    • Peek at the last digit. A perfect square can only end in 0, 1, 4, 5, 6 or 9. If your number ends in 2, 3, 7 or 8, don’t expect a whole-number answer.
    • Count the pairs. The number of pairs tells you how many digits the whole-number part of the root will have.
    • Keep your columns neat. Honestly, most mistakes come from messy layout, not from the maths.

    Mistakes worth avoiding

    • Pairing from the wrong side. Whole numbers are paired from the right. Decimals are paired outwards from the decimal point, in both directions.
    • Doubling only the last digit. You double the whole answer so far, not just the digit you found last.
    • Picking a digit that’s too big. If your product overshoots the dividend, drop the digit by one and try again.
    • Bringing down single zeros. When you add zeros for decimal places, bring them down in pairs.
    • Putting the decimal point in the wrong place. It goes into the answer the moment you bring down your first decimal pair.
    • Skipping the check. Ten seconds at the end can save you a lost mark, as you’re about to see.

    How to check your answer

    For a perfect square, multiply your answer by itself and see if you get the original number back. 23 × 23 = 529, so we’re good.

    For a non-perfect square, square your approximation and see how close it lands. 8.426² comes out at about 70.997, which is very nearly 71. Carry the division further and it gets closer still.

    Read More:

    1. Application for Leave of Absence

    2. 132000 in Words

    When should you actually use this method?

    Situation

    Best method

    Small perfect squares (up to about 400)

    Memorised squares or prime factorisation

    Large perfect squares

    Long division

    Non-perfect squares

    Long division

    Decimal numbers

    Long division

    A quick rough answer

    Estimating between two perfect squares

    Everyday life, no rules

    A calculator, obviously

    Calculators are great outside the classroom. But in plenty of school exams, especially in the middle and early high school years, you’re expected to show your working. That’s exactly where this method comes in.

    Frequently Asked Questions

     It’s a digit-by-digit technique for finding a square root. You pair up the digits, then repeatedly divide, multiply, subtract and bring down the next pair.

    Because squaring a one-digit number gives at most two digits, and squaring a two-digit number gives at most four. So every pair of digits in the number lines up with exactly one digit in the root.

    Add pairs of zeros after the decimal point and keep going until you’ve got the accuracy you need. The remainder will never reach zero.

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