Finding a square root by the long division method takes many steps. When you already know the number is a perfect square, there is a much faster way. It reads the answer straight off the number: the first digit of the root comes from the front of the number, and the last digit comes from the end. This guide shows the method for four-digit squares, extends it to five and six digits, explains how to tell whether a number is a perfect square at all, and ends with a practice set.
A perfect square is a whole number multiplied by itself. For example, 32 × 32 = 1024, so 1024 is a perfect square and its square root, √1024, is 32. The method depends on knowing the squares from 1 to 9 well, and our squares 1 to 20 list is a good place to learn them.
Mark the digits in pairs, starting from the right. Each pair gives one digit of the root, so the number of pairs tells you how many digits the answer has.
Look at the leftmost pair and find the largest number whose square fits into it. For 1024, the leftmost pair is 10. The largest square that fits is 9, which is 3 squared, so the root starts with 3.
The last digit of a square depends only on the last digit of its root. So the last digit of the number narrows the last digit of the root to two choices:
| The number ends in | The root ends in |
|---|---|
| 1 | 1 or 9 |
| 4 | 2 or 8 |
| 9 | 3 or 7 |
| 6 | 4 or 6 |
| 5 | 5 |
| 0 | 0 |
A perfect square never ends in 2, 3, 7 or 8. 1024 ends in 4, so its root ends in 2 or 8: it is either 32 or 38.
Take the first digit you found in Step 2 and multiply it by one more than itself. For 1024, that is 3 × 4 = 12. Then compare with the leftmost part of the number:
The leftmost part of 1024 is 10, which is smaller than 12, so the root ends in the smaller choice, 2. So √1024 is 32.
Why it works: 3 × 4 = 12 is the front of 35², which is 1,225. Any number below 1,225 has a root below 35, and any number above it has a root above 35. So the comparison is really asking whether the number is below or above the square of the 5 in the middle.
Some books describe the same test with a remainder: take the largest square away from the leftmost part (10 − 9 = 1), and if that is less than the first digit of the root, choose the smaller ending. It gives the same answer, since both ask whether the number is below 35².
√2209. The leftmost pair is 22. The largest square that fits is 16, from 4, so the root starts with 4. The number ends in 9, so the root ends in 3 or 7. Then 4 × 5 = 20, and 22 is bigger, so choose 7. The root is 47.
√6889. The leftmost pair is 68, and 64 is the largest square that fits, so the root starts with 8. It ends in 9, so the root ends in 3 or 7. Then 8 × 9 = 72, and 68 is smaller, so choose 3. The root is 83.
√9801. The leftmost pair is 98, so the root starts with 9. It ends in 1, so the root ends in 1 or 9. Then 9 × 10 = 90, and 98 is bigger, so choose 9. The root is 99.
√3136. The leftmost pair is 31, so the root starts with 5. It ends in 6, so the root ends in 4 or 6. Then 5 × 6 = 30, and 31 is bigger, so choose 6. The root is 56.
For bigger numbers, use everything in front of the last pair as the leftmost part. Its square root gives all the digits of the answer except the last.
√15129. Everything in front of the last pair is 151. The largest square that fits is 144, from 12, so the root starts with 12. It ends in 9, so the root is 123 or 127. Then 12 × 13 = 156, and 151 is smaller, so choose 3. The root is 123.
√75076. In front of the last pair is 750. Since 27 × 27 = 729 and 28 × 28 = 784, the root starts with 27. It ends in 6, so the root is 274 or 276. Then 27 × 28 = 756, and 750 is smaller, so choose 4. The root is 274.
The method only works for perfect squares. On any other number it gives a wrong answer that looks believable. Two quick tests rule out most numbers:
If the number passes both tests, find the root and then square it to make sure. For numbers that aren't perfect squares, such as 70 or 85, our guide to fast square root calculations shows how to estimate.
Square the root and compare. The Vedic Maths square tricks make this quick. For 47, the duplex columns are 16, 56 and 49. Write 9 and carry 4, then 56 + 4 = 60, so write 0 and carry 6, then 16 + 6 = 22. That gives 2,209, so √2209 is 47.
Answers: (1) 36: it starts with 3, ends in 4 or 6, and 3 × 4 = 12 is the same as the leftmost pair, so choose the bigger ending. (2) 74: it starts with 7, ends in 4 or 6, and 54 is smaller than 7 × 8 = 56. (3) 89: it starts with 8, ends in 1 or 9, and 79 is bigger than 8 × 9 = 72. (4) 57: it starts with 5, ends in 3 or 7, and 32 is bigger than 5 × 6 = 30. (5) 141: the front is 198, whose largest square is 196 from 14, it ends in 1 or 9, and 198 is smaller than 14 × 15 = 210. (6) No: it ends in 8.
Square roots are only quick when squares are, so keep the facts behind them fresh with the multiplication game and the times tables chart. For more methods like this one, see all our Vedic Maths guides.
No. It only gives the right answer for perfect squares. Check first that the number ends in 0, 1, 4, 5, 6 or 9 and that its digital root is 1, 4, 7 or 9, and always square your answer to confirm.
Multiply the first part of the root by one more than itself. If the front of the number is smaller than that, choose the smaller ending. If it is the same or bigger, choose the bigger one.
Yes. Cube roots of perfect cubes can be read off in a similar way, using the last digit and the front of the number. See our Vedic Maths cube root tricks.