A square root answers one question: which number, multiplied by itself, gives this one? For 144 the answer is 12, because 12 × 12 = 144. Finding square roots quickly comes down to two skills: recognising perfect squares, and estimating sensibly when the number isn't one. This guide covers both, plus how to simplify roots like √300, with checked examples and a practice set.
A perfect square is a whole number times itself. Knowing the first twenty by heart makes every other method faster, because they are the landmarks you estimate from.
So √144 is 12, √196 is 14, and √400 is 20. If you haven't learned them yet, our list of squares from 1 to 20 has the full table and memory tips. Two patterns help: every square of a number ending in 0 ends in 00, and every square of a number ending in 5 ends in 25.
For a four-digit perfect square, you can find the root in two quick steps, using its size and its last digit.
Step 1: the tens digit. Look at the number without its last two digits, and find the largest square that fits. For 2,304, that is 23, and the largest square under 23 is 16, which is 4 squared. So the root is in the 40s.
Step 2: the ones digit. The last digit of a square depends only on the last digit of the root:
| Square ends in | Root ends in |
|---|---|
| 1 | 1 or 9 |
| 4 | 2 or 8 |
| 9 | 3 or 7 |
| 6 | 4 or 6 |
| 5 | 5 |
| 0 | 0 |
2,304 ends in 4, so the root is 42 or 48. To choose, compare with 45 squared, which is 2,025. 2,304 is bigger, so the root is the larger option: 48. Check: 48 × 48 = 2304.
Another one: 5,329. The first part is 53, and the largest square under 53 is 49, so the root is in the 70s. It ends in 9, so the root is 73 or 77. 75 squared is 5,625, which is bigger than 5,329, so the root is 73. Check: 73 × 73 = 5329.
The Vedic Maths version of this method is in square roots of perfect squares.
Most numbers aren't perfect squares, so their roots aren't whole numbers. Start by finding the two perfect squares either side. For √70, those are 64 and 81, so √70 is between 8 and 9. Since 70 is closer to 64, the root is closer to 8.
To get one decimal place, use this rule: take the nearest square root below, then add how far past that square you are, divided by twice that root. For √70, you are 6 past 64, and twice 8 is 16. So √70 is about 8 + 6/16, which is 8.375. The true value is about 8.367, so the estimate is very close.
A few more with the same rule:
The rule slightly overestimates, more so when the number is far from the square below. For a closer answer, use the method in the next section.
Newton's method turns any reasonable guess into a better one. Divide the number by your guess, then average the result with your guess.
For √85, start with an initial guess of 9. Average the guess with 85 divided by the guess: (9 + 85/9) / 2 = 83/9 ≈ 9.22. Check: 9.22 × 9.22 ≈ 85.01, so √85 ≈ 9.22. Repeat the step with 9.22 as the new guess if you need more accuracy.
Why it works: if your guess is too small, the number divided by it is too big, and the true root lies between them. Averaging lands close to it.
In algebra and geometry, you often keep an exact root instead of a decimal. To simplify, pull out the largest square factor.
Square it. If your root times itself is close to the original number, you are right. For estimates, also check it sits between the two whole numbers you started from: √70 must be between 8 and 9, so an answer like 7.4 or 9.1 is a slip.
Answers: (1) 14. (2) 32: 10 lies between 9 and 16, so the root is in the 30s; 1,024 ends in 4, so it is 32 or 38, and 35 squared is 1,225, too big. (3) 59: 34 lies between 25 and 36, so the root is in the 50s; it ends in 1, so it is 51 or 59, and 55 squared is 3,025, which is smaller, so 59. (4) About 7.07. (5) 4√3. (6) About 10.5.
Square roots are only as quick as the squares behind them. Our guide to squaring numbers fast builds that skill, the times tables chart covers the squares up to 12 × 12, and the multiplication game keeps them sharp.
Find the perfect squares on either side to get the whole-number part. Then estimate the decimal by dividing how far you are past the lower square by twice its root, or improve a guess with Newton's method.
Check the last digit first: a perfect square can only end in 0, 1, 4, 5, 6 or 9. If it passes, find the nearest roots and square them. If neither gives the number exactly, it isn't a perfect square.
It is the exact value in its simplest form. 10√3 is easier to work with in algebra, and you can still turn it into a decimal (about 17.32) when you need one.