√966 at a glance
- Exact value
- √966
- Decimal (10 places)
- 31.0805405358
- Rounded
- 31.1 · 31.08 · 31.081
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.080541
- Prime factorization
- 2 × 3 × 7 × 23
- Cube root
- 9.885357
How to simplify √966
The prime factorization of 966 is 2 × 3 × 7 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √966 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 966, 2, 3, 7 and 23 appear an odd number of times, so √966 is irrational and 31.0805405358 is a rounded value.
Where √966 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √966 lies between 31 and 32. 966 is 5 above 961 and 58 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.0794 (0% low)
- Tangent from 31, i.e. 31 + 5 ÷ 62: 31.0806 (0% high)
- Tangent from 32, i.e. 32 − 58 ÷ 64: 31.0938 (0.04% high)
For √966 the tangent at 31 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 966 is just 5 above 961.
Finding √966 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, 31 (31² = 961):
| Step | Guess x | 966 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.1612903226 | 31.0806451613 | 3 |
| 2 | 31.0806451613 | 31.0804359107 | 31.0805405360 | 9 |
| 3 | 31.0805405360 | 31.0805405357 | 31.0805405358 | 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 √966 = 31.0805405358 to every decimal shown.
√966 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √966 the pattern is [31; 12, 2, 2, 2, 12, 62] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √966 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 |
|---|---|---|
| 31/1 | 31.0000000000 | 8.1 × 10⁻² |
| 373/12 | 31.0833333333 | 2.8 × 10⁻³ |
| 777/25 | 31.0800000000 | 5.4 × 10⁻⁴ |
| 1,927/62 | 31.0806451613 | 1.0 × 10⁻⁴ |
| 4,631/149 | 31.0805369128 | 3.6 × 10⁻⁶ |
| 57,499/1,850 | 31.0805405405 | 4.7 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 966y² = 1. Its smallest solution in positive whole numbers is x = 57,499, y = 1,850.
√966 in geometry and everyday measurements
- 966 square feet is 89.7 m². Laid out as a square — a small house footprint or a lot — it is about 31.08 ft (31 ft 1 in) on a side.
- 966 is not a sum of two whole-number squares — the prime factor 3, 7 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √966 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 31 box, because 1² + 2² + 31² = 966.
Square roots near √966 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √963 | 3√107 | 31.0322 | No |
| √964 | 2√241 | 31.0483 | No |
| √965 | √965 | 31.0644 | No |
| √966 | √966 | 31.0805 | No |
| √967 | √967 | 31.0966 | No |
| √968 | 22√2 | 31.1127 | No |
| √969 | √969 | 31.1288 | No |
- The cube root of 966 is about 9.885357.
- Squaring undoes the root: (√966)² = 966, while 966² = 933,156 — the number whose square root is 966.
Frequently asked questions
What is the square root of 966?
The square root of 966 is √966, about 31.0805405358. The negative root, −31.080541, also squares to 966.
Is the square root of 966 rational or irrational?
Irrational. 966 is not a perfect square — it falls between 961 and 1024 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √966 be simplified?
No. 966 = 2 × 3 × 7 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √966 rounded to two decimal places?
√966 ≈ 31.08 to two decimal places (31.1 to one, 31.081 to three). Check: 31.08² = 965.9664, close to 966.