√129 at a glance
- Exact value
- √129
- Decimal (10 places)
- 11.3578166916
- Rounded
- 11.4 · 11.36 · 11.358
- Perfect square?
- No — between 11² and 12²
- Rational?
- Irrational
- Both square roots
- ±11.357817
- Prime factorization
- 3 × 43
- Cube root
- 5.052774
How to simplify √129
The prime factorization of 129 is 3 × 43. Every prime appears only once, so there is no pair to bring outside the radical — √129 is already in simplest form (a "square-free" radicand).
A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 129, 3 and 43 appear an odd number of times, so √129 is irrational and 11.3578166916 is a rounded value.
Where √129 sits between perfect squares
121 = 11² and 144 = 12² are the nearest perfect squares, so √129 lies between 11 and 12. 129 is 8 above 121 and 15 below 144, so the root is closer to 11.
- Straight line between 121 and 144: 11.3478 (0.09% low)
- Tangent from 11, i.e. 11 + 8 ÷ 22: 11.3636 (0.05% high)
- Tangent from 12, i.e. 12 − 15 ÷ 24: 11.3750 (0.15% high)
For √129 the tangent at 11 wins, missing by only 0.0058. Tangent estimates shine when the number sits close to a perfect square — here 129 is just 8 above 121.
Finding √129 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 129: following the tangent line down to zero simplifies to averaging x with 129 ÷ x.
Start from the nearest whole number, 11 (11² = 121):
| Step | Guess x | 129 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 11.0000000000 | 11.7272727273 | 11.3636363636 | 2 |
| 2 | 11.3636363636 | 11.3520000000 | 11.3578181818 | 5 |
| 3 | 11.3578181818 | 11.3578152014 | 11.3578166916 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √129 = 11.3578166916 to every decimal shown.
√129 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √129 the pattern is [11; 2, 1, 3, 1, 6, 1, 3, 1, 2, 22] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √129 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 | 3.6 × 10⁻¹ |
| 23/2 | 11.5000000000 | 1.4 × 10⁻¹ |
| 34/3 | 11.3333333333 | 2.4 × 10⁻² |
| 125/11 | 11.3636363636 | 5.8 × 10⁻³ |
| 159/14 | 11.3571428571 | 6.7 × 10⁻⁴ |
| 1,079/95 | 11.3578947368 | 7.8 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 129y² = 1. Its smallest solution in positive whole numbers is x = 16,855, y = 1,484.
√129 in geometry and everyday measurements
- A square room or garden bed covering 129 square feet measures about 11.36 ft (11 ft 4 in) along each wall.
- 129 is not a sum of two whole-number squares — the prime factor 3 and 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √129 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 8 × 8 box, because 1² + 8² + 8² = 129.
Square roots near √129 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √126 | 3√14 | 11.2250 | No |
| √127 | √127 | 11.2694 | No |
| √128 | 8√2 | 11.3137 | No |
| √129 | √129 | 11.3578 | No |
| √130 | √130 | 11.4018 | No |
| √131 | √131 | 11.4455 | No |
| √132 | 2√33 | 11.4891 | No |
- The cube root of 129 is about 5.052774.
- Four times the radicand doubles the root: √516 = 2 × √129 ≈ 22.715633.
Frequently asked questions
What is the square root of 129?
The square root of 129 is √129, about 11.3578166916. The negative root, −11.357817, also squares to 129.
Is the square root of 129 rational or irrational?
Irrational. 129 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 √129 be simplified?
No. 129 = 3 × 43 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √129 rounded to two decimal places?
√129 ≈ 11.36 to two decimal places (11.4 to one, 11.358 to three). Check: 11.36² = 129.0496, close to 129.