Long Division Method for Square Root: Step-by-Step Guide with Solved Examples

The long division method for square root is a way to find a square root digit by digit, with no calculator. It works on perfect squares, decimals, and even numbers like 2 or 5 that never come out neat.

Here’s the whole idea in five lines:

Pair the digits, starting from the decimal point.
Find the biggest square that fits into the first pair. That’s the first digit of your answer.
Subtract, then bring down the next pair.
Double your answer so far and find the next digit that fits.
Repeat until the remainder is zero or you have enough decimal places.

It looks fiddly the first time, but it’s very mechanical once you’ve done a couple. I’ll go through each step slowly, then solve five examples so you can see the pattern for yourself.

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Why Use the Long Division Method for Square Root ?

Fair question, since every phone has a calculator. But there are a few good reasons:

Exams still ask for it. Many school and competitive papers say “find the square root by long division,” and the marks are in the working.
It works on any number. Prime factorization only helps with perfect squares. Long division also handles 2, 5, or 152.2756.
You choose the accuracy. Want five decimal places? Keep adding pairs of zeros and keep going.
It builds number sense. You start to feel how digits, squares and place value connect.

Most students who get stuck on this aren’t struggling with the math. It’s usually one small slip, like pairing the digits from the wrong side or forgetting to double the quotient. Sort those out and the rest falls into place.

What You Need to Know Before Using the Long Division Method for Square Root

Perfect squares. A perfect square is a number like 49 or 144, which comes from multiplying a whole number by itself. If you know your squares up to about 20, you’ll save a lot of time.
Why do we pair digits? Squaring a number roughly doubles its digit count, so square roots work in pairs of digits:

529 has 3 digits, so its root has 2 digits
2916 has 4 digits, so its root has 2 digits
6561 has 4 digits, so its root has 2 digits

The rule: a number with n digits has a square root with ⌈n/2⌉ digits (n divided by 2, rounded up).

The divisor trick. At each step, you double the answer so far and add a new digit to the end of it. Then you multiply by that same new digit. That feels strange at first, and the examples below will make it click.

The Long Division Method for Square Root, Step by Step

Here’s the full process.

Step 1: Group the digits in pairs.
For the whole-number part, start at the right and move left. For the decimal part, start at the decimal point and move right. If the last decimal pair has only one digit, add a zero. If the leftmost group is a single digit, that’s fine.

Examples: 2916 → 29 | 16. And 152.2756 → 1 | 52 . 27 | 56.

Step 2: Find the first digit.
Look at the leftmost group. Find the largest number whose square fits inside it. Write that number as the first digit of the answer (the quotient) and as your first divisor. Subtract the square.

Step 3: Bring down the next pair.
Place it next to the remainder. That’s your new dividend.

Step 4: Double the quotient.
Write the doubled quotient as your new divisor, with a blank at the end (like 4_).

Step 5: Find the missing digit.
Pick the largest digit d so that (divisor with d attached) × d fits into the dividend. Write d in the blank and also in the quotient.

Step 6: Subtract and repeat.
Subtract, bring down the next pair, double the whole quotient again, and go around once more.

Step 7: Deal with decimals.
When you bring down the first decimal pair, put a decimal point in the quotient. If the number isn’t a perfect square, keep adding pairs of zeros (00) for more decimal places.

Solved Example 1: Square Root of 529

Pairs: 5 | 29.

The largest square that fits into 5 is 4 (since 2² = 4). The quotient starts with 2. Remainder: 5 − 4 = 1.
Bring down 29 to get 129.
Double the quotient: 2 × 2 = 4. The new divisor is 4_.
Find d so that (40 + d) × d ≤ 129. Try d = 3: 43 × 3 = 129. A perfect fit.
Remainder: 0. Done.

√529 = 23

Quick check: 23 × 23 = 529. ✓

Solved Example 2: Square Root of 2916

Pairs: 29 | 16.

The largest square that fits into 29 is 25 (5²). Quotient: 5. Remainder: 29 − 25 = 4.
Bring down 16 to get 416.
Double the quotient: 5 × 2 = 10. Find d so that (100 + d) × d ≤ 416.
Try d = 4: 104 × 4 = 416. Perfect.
Remainder: 0.

√2916 = 54

Quick check: 54 × 54 = 2916. ✓

Solved Example 3: Square Root of 2 (a Non-Perfect Square)

This is where the method really shines. Let’s find √2 to three decimal places. Write 2 as 2.00 00 00.

Step	Working	Quotient so far
1	The largest square that fits into 2 is 1. Subtract: 2 − 1 = 1	1
2	Bring down 00 → 100. Double 1 = 2. Find d: (20 + d) × d ≤ 100. d = 4 gives 24 × 4 = 96. Remainder 4	1.4
3	Bring down 00 → 400. Double 14 = 28. (280 + d) × d ≤ 400. d = 1 gives 281 × 1 = 281. Remainder 119	1.41
4	Bring down 00 → 11900. Double 141 = 282. (2820 + d) × d ≤ 11900. d = 4 gives 2824 × 4 = 11296. Remainder 604	1.414

√2 ≈ 1.414

The true value is 1.41421356…, so we’re right on target. Want more decimal places? Bring down another 00 and keep going.

Solved Example 4: Square Root of a Decimal (152.2756)

Pairs: 1 | 52 . 27 | 56. The whole-number part has three digits, so the first group is just “1”.

The largest square that fits into 1 is 1. Quotient: 1. Remainder: 0.
Bring down 52 → 52. Double 1 = 2. Find d: (20 + d) × d ≤ 52. d = 2 gives 22 × 2 = 44. Remainder 8. Quotient: 12.
Bring down 27. We’ve now crossed the decimal point, so put a point in the quotient. Dividend: 827. Double 12 = 24. (240 + d) × d ≤ 827. d = 3 gives 243 × 3 = 729. Remainder 98. Quotient: 12.3.
Bring down 56 → 9856. Double 123 = 246. (2460 + d) × d ≤ 9856. d = 4 gives 2464 × 4 = 9856. Remainder 0. Quotient: 12.34.

√152.2756 = 12.34

Quick check: 12.34 × 12.34 = 152.2756. ✓

Solved Example 5: Square Root of 5 (to Two Decimal Places)

Write 5 as 5.00 00.

The largest square that fits into 5 is 4. Quotient: 2. Remainder: 1.
Bring down 00 → 100. Double 2 = 4. (40 + d) × d ≤ 100. d = 2 gives 42 × 2 = 84. Remainder 16. Quotient: 2.2.
Bring down 00 → 1600. Double 22 = 44. (440 + d) × d ≤ 1600. d = 3 gives 443 × 3 = 1329. (d = 4 would give 1776, which is too big.) Remainder 271. Quotient: 2.23.

√5 ≈ 2.23 (more precisely, 2.236…)

Where to Place the Decimal Point in the Long Division Method for Square Root

This trips up a lot of people, so here’s the simple rule: the decimal point goes in the quotient the moment you bring down the first decimal pair. Not before, and not after.

If your number has no decimal point (like 2 or 5) and you want decimals, add the point and pairs of zeros yourself. The moment you bring down the first “00”, drop the decimal point into your answer.

Mistakes Almost Everyone Makes

Pairing from the wrong side. For whole numbers, pair from the right. For decimals, pair from the decimal point going right. Pairing 2916 as 2 | 91 | 6 will wreck everything.
Forgetting to double the quotient. Double the whole quotient so far, not just the last digit.
Using different digits in different places. The digit you attach to the divisor, the digit you multiply by, and the digit you write in the quotient must all be the same.
Picking a digit that’s too big. If the product is larger than the dividend, step back one digit. If a bigger digit would still have fit, you went too small.
Leaving a lone decimal digit unpaired. In 12.5, write it as 12.50 so “50” forms a proper pair.
Misplacing the decimal point. Add it to the quotient right when you bring down the first decimal pair.

Long Division vs Other Square Root Methods

Method	Perfect squares	Non-perfect squares	Speed
Prime factorization	Yes	No	Fast for small numbers
Estimation / guess and check	Roughly	Roughly	Quick but approximate
Long division method	Yes	Yes, to any decimal place	Steady and reliable
Calculator	Yes	Yes	Instant, but no working shown

Prime factorization is quicker for something like 144, but it falls apart on 2 or 152.2756. Long division treats them all the same way, which is why it’s worth learning.

Tips and Tricks to Master the Long Division Method

Memorize squares up to 20 or 25. It makes the first step instant.
Use the last digit as a clue. A perfect square ending in 9 has a root ending in 3 or 7. One ending in 6 has a root ending in 4 or 6.
Check your answer by squaring it. It takes ten seconds and catches most errors.
Start easy. Try perfect squares first (like 484, 1024 or 6561), then decimals, then non-perfect squares.
Write neatly. Many of the mistakes people make here are alignment slips, not math errors. Keep your columns tidy.

Conclusion

- Pair the digits, starting from the decimal point
- Find the largest square that fits the first pair
- Bring down the next pair, double the quotient, and find the next digit
- Put a decimal point in the quotient when you bring down the first decimal pair
- For non-perfect squares, keep adding pairs of zeros
- Always check your answer by squaring it

Want some practice? Try √1156 (answer: 34), √7569 (answer: 87), and √3 to two decimal places (answer: 1.73). Do those three and the method will stick for good.

Frequently Asked Questions

What is the long division method for square root?

It’s a step-by-step process that finds a square root one digit at a time. You pair the digits, find the largest square that fits the first pair, then keep doubling the quotient to find each next digit.

Can I use it for decimal numbers?

Yes. Pair the decimal digits starting from the decimal point, and put the decimal point in the answer when you bring down the first decimal pair.

How do I find the square root of a non-perfect square this way?

Add a decimal point and pairs of zeros (00) after the number, and keep dividing. Each new pair gives you one more decimal place.

Why do we pair the digits?

Because squaring a number roughly doubles its digit count. Each pair of digits in the number matches one digit in the root.

Why do we double the quotient at each step?

It comes from the identity (a + b)² = a² + 2ab + b². Doubling the quotient gives you the 2a part, which lets you find the next digit b.