√126 at a glance
- Exact value
- 3√14
- Decimal (10 places)
- 11.2249721603
- Rounded
- 11.2 · 11.22 · 11.225
- Perfect square?
- No — between 11² and 12²
- Rational?
- Irrational
- Both square roots
- ±11.224972
- Prime factorization
- 2 × 3² × 7
- Cube root
- 5.013298
How to simplify √126
Look for the largest perfect square that divides 126. Here it is 9 (3²), because 126 = 9 × 14 and 14 has no square factor left:
The prime factorization tells the same story: 126 = 2 × 3² × 7. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 7 stays inside.
Check: (3√14)² = 3² × 14 = 9 × 14 = 126. As a decimal, 3√14 = 3 × 3.7416573868 ≈ 11.2249721603.
Where √126 sits between perfect squares
121 = 11² and 144 = 12² are the nearest perfect squares, so √126 lies between 11 and 12. 126 is 5 above 121 and 18 below 144, so the root is closer to 11.
- Straight line between 121 and 144: 11.2174 (0.07% low)
- Tangent from 11, i.e. 11 + 5 ÷ 22: 11.2273 (0.02% high)
- Tangent from 12, i.e. 12 − 18 ÷ 24: 11.2500 (0.22% high)
For √126 the tangent at 11 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 126 is just 5 above 121.
Finding √126 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 11 (11² = 121):
| Step | Guess x | 126 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 11.0000000000 | 11.4545454545 | 11.2272727273 | 2 |
| 2 | 11.2272727273 | 11.2226720648 | 11.2249723960 | 6 |
| 3 | 11.2249723960 | 11.2249719246 | 11.2249721603 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √126 = 11.2249721603 to every decimal shown.
√126 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √126 the pattern is [11; 4, 2, 4, 22] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √126 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 11/1 | 11.0000000000 | 2.2 × 10⁻¹ |
| 45/4 | 11.2500000000 | 2.5 × 10⁻² |
| 101/9 | 11.2222222222 | 2.7 × 10⁻³ |
| 449/40 | 11.2250000000 | 2.8 × 10⁻⁵ |
| 9,979/889 | 11.2249718785 | 2.8 × 10⁻⁷ |
| 40,365/3,596 | 11.2249721913 | 3.1 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 126y² = 1. Its smallest solution in positive whole numbers is x = 449, y = 40.
√126 in geometry and everyday measurements
- A square room or garden bed covering 126 square feet measures about 11.22 ft (11 ft 3 in) along each wall.
- 126 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √126 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 11 box, because 1² + 2² + 11² = 126.
- Since √126 = 3√14, a length of √126 is exactly 3 copies of the length √14 laid end to end.
Square roots near √126 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √123 | √123 | 11.0905 | No |
| √124 | 2√31 | 11.1355 | No |
| √125 | 5√5 | 11.1803 | No |
| √126 | 3√14 | 11.2250 | No |
| √127 | √127 | 11.2694 | No |
| √128 | 8√2 | 11.3137 | No |
| √129 | √129 | 11.3578 | No |
- The cube root of 126 is about 5.013298.
- Four times the radicand doubles the root: √504 = 2 × √126 ≈ 22.449944.
Frequently asked questions
What is the square root of 126?
The square root of 126 is 3√14 in simplest radical form, which is about 11.2249721603. The negative root, −11.224972, also squares to 126.
Is the square root of 126 rational or irrational?
Irrational. 126 is not a perfect square — it falls between 121 and 144 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √126 be simplified?
Yes. The largest perfect square dividing 126 is 9, so √126 = √9 × √14 = 3√14.
What is √126 rounded to two decimal places?
√126 ≈ 11.22 to two decimal places (11.2 to one, 11.225 to three). Check: 11.22² = 125.8884, close to 126.