√906 at a glance
- Exact value
- √906
- Decimal (10 places)
- 30.0998338866
- Rounded
- 30.1 · 30.10 · 30.100
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.099834
- Prime factorization
- 2 × 3 × 151
- Cube root
- 9.676302
How to simplify √906
The prime factorization of 906 is 2 × 3 × 151. Every prime appears only once, so there is no pair to bring outside the radical — √906 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 906, 2, 3 and 151 appear an odd number of times, so √906 is irrational and 30.0998338866 is a rounded value.
Where √906 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √906 lies between 30 and 31. 906 is 6 above 900 and 55 below 961, so the root is closer to 30.
- Straight line between 900 and 961: 30.0984 (0% low)
- Tangent from 30, i.e. 30 + 6 ÷ 60: 30.1000 (0% high)
- Tangent from 31, i.e. 31 − 55 ÷ 62: 30.1129 (0.04% high)
For √906 the tangent at 30 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 906 is just 6 above 900.
Finding √906 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, 30 (30² = 900):
| Step | Guess x | 906 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 30.2000000000 | 30.1000000000 | 3 |
| 2 | 30.1000000000 | 30.0996677741 | 30.0998338870 | 9 |
| 3 | 30.0998338870 | 30.0998338861 | 30.0998338866 | all 10 shown |
The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √906 = 30.0998338866 to every decimal shown.
√906 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √906 the pattern is [30; 10, 60] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √906 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 |
|---|---|---|
| 30/1 | 30.0000000000 | 1.0 × 10⁻¹ |
| 301/10 | 30.1000000000 | 1.7 × 10⁻⁴ |
| 18,090/601 | 30.0998336106 | 2.8 × 10⁻⁷ |
| 181,201/6,020 | 30.0998338870 | 4.6 × 10⁻¹⁰ |
| 10,890,150/361,801 | 30.0998338866 | < 10⁻¹⁰ |
| 109,082,701/3,624,030 | 30.0998338866 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 906y² = 1. Its smallest solution in positive whole numbers is x = 301, y = 10.
√906 in geometry and everyday measurements
- 906 square feet is 84.2 m². Laid out as a square — a small house footprint or a lot — it is about 30.1 ft (30 ft 1 in) on a side.
- 906 is not a sum of two whole-number squares — the prime factor 3 and 151 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √906 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 8 × 29 box, because 1² + 8² + 29² = 906.
Square roots near √906 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √903 | √903 | 30.0500 | No |
| √904 | 2√226 | 30.0666 | No |
| √905 | √905 | 30.0832 | No |
| √906 | √906 | 30.0998 | No |
| √907 | √907 | 30.1164 | No |
| √908 | 2√227 | 30.1330 | No |
| √909 | 3√101 | 30.1496 | No |
- The cube root of 906 is about 9.676302.
- Squaring undoes the root: (√906)² = 906, while 906² = 820,836 — the number whose square root is 906.
Frequently asked questions
What is the square root of 906?
The square root of 906 is √906, about 30.0998338866. The negative root, −30.099834, also squares to 906.
Is the square root of 906 rational or irrational?
Irrational. 906 is not a perfect square — it falls between 900 and 961 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √906 be simplified?
No. 906 = 2 × 3 × 151 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √906 rounded to two decimal places?
√906 ≈ 30.10 to two decimal places (30.1 to one, 30.100 to three). Check: 30.10² = 906.01, close to 906.