Simplest radical form
√12 = √(4 × 3) = 2√3
Long division method
- Group the digits in pairs from the decimal point: 12. The largest square ≤ 12 is 3² = 9, so the first digit is 3; remainder 3.
- Bring down 00: 300. Double the root so far (3 → 6) and find the largest d with (6d) × d ≤ 300: d = 4, since 64 × 4 = 256. Remainder 44.
- Bring down 00: 4400. Double the root so far (34 → 68) and find the largest d with (68d) × d ≤ 4400: d = 6, since 686 × 6 = 4116. Remainder 284.
- Bring down 00: 28400. Double the root so far (346 → 692) and find the largest d with (692d) × d ≤ 28400: d = 4, since 6924 × 4 = 27696. Remainder 704.
The digits read 3.464, so √12 ≈ 3.464.
Quick estimate
3² = 9 and 4² = 16, so √12 lies between 3 and 4. A fast first guess is 3 + 3/(2 × 3) = 3.500, against the true 3.464.
Square roots of other numbers
| Root | Value |
|---|---|
| √2 | 1.4142 |
| √3 | 1.7321 |
| √5 | 2.2361 |
| √6 | 2.4495 |
| √7 | 2.6458 |
| √8 | 2.8284 |
| √10 | 3.1623 |
| √11 | 3.3166 |
| √12 | 3.4641 |
| √13 | 3.6056 |
| √14 | 3.7417 |
| √15 | 3.8730 |
| √17 | 4.1231 |
| √18 | 4.2426 |
| √19 | 4.3589 |
| √20 | 4.4721 |
| √21 | 4.5826 |
| √22 | 4.6904 |
| √23 | 4.7958 |
| √24 | 4.8990 |
| √26 | 5.0990 |
| √27 | 5.1962 |
| √28 | 5.2915 |
| √29 | 5.3852 |
| √30 | 5.4772 |
| √31 | 5.5678 |
| √32 | 5.6569 |
| √33 | 5.7446 |
| √34 | 5.8310 |
| √35 | 5.9161 |
| √37 | 6.0828 |
| √38 | 6.1644 |
| √39 | 6.2450 |
| √40 | 6.3246 |
| √41 | 6.4031 |
| √42 | 6.4807 |
| √43 | 6.5574 |
| √44 | 6.6332 |
| √45 | 6.7082 |
| √46 | 6.7823 |
| √47 | 6.8557 |
| √48 | 6.9282 |
| √50 | 7.0711 |
| √72 | 8.4853 |
| √75 | 8.6603 |
| √80 | 8.9443 |
| √98 | 9.8995 |
| √125 | 11.1803 |
| √200 | 14.1421 |
Questions students ask
What is the square root of 12?
√12 ≈ 3.4641. It is irrational, so the decimal never ends or repeats.
Is √12 rational?
No. 12 is not a perfect square, so √12 is irrational.
What is √12 in simplest radical form?
√12 = √(4 × 3) = 2√3
How do you find √12 by long division?
Pair the digits, find the largest square under the first pair, then repeatedly double the root so far and find the next digit. It gives √12 ≈ 3.464.