√982 at a glance
- Exact value
- √982
- Decimal (10 places)
- 31.3368792320
- Rounded
- 31.3 · 31.34 · 31.337
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.336879
- Prime factorization
- 2 × 491
- Cube root
- 9.939636
How to simplify √982
The prime factorization of 982 is 2 × 491. Every prime appears only once, so there is no pair to bring outside the radical — √982 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 982, 2 and 491 appear an odd number of times, so √982 is irrational and 31.3368792320 is a rounded value.
Where √982 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √982 lies between 31 and 32. 982 is 21 above 961 and 42 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.3333 (0.01% low)
- Tangent from 31, i.e. 31 + 21 ÷ 62: 31.3387 (0.01% high)
- Tangent from 32, i.e. 32 − 42 ÷ 64: 31.3438 (0.02% high)
For √982 the tangent at 31 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 982 is just 21 above 961.
Finding √982 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 | 982 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.6774193548 | 31.3387096774 | 2 |
| 2 | 31.3387096774 | 31.3350488935 | 31.3368792854 | 7 |
| 3 | 31.3368792854 | 31.3368791785 | 31.3368792320 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √982 = 31.3368792320 to every decimal shown.
√982 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √982 the pattern is [31; 2, 1, 30, 1, 2, 62] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √982 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 | 3.4 × 10⁻¹ |
| 63/2 | 31.5000000000 | 1.6 × 10⁻¹ |
| 94/3 | 31.3333333333 | 3.5 × 10⁻³ |
| 2,883/92 | 31.3369565217 | 7.7 × 10⁻⁵ |
| 2,977/95 | 31.3368421053 | 3.7 × 10⁻⁵ |
| 8,837/282 | 31.3368794326 | 2.0 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 982y² = 1. Its smallest solution in positive whole numbers is x = 8,837, y = 282.
√982 in geometry and everyday measurements
- 982 square feet is 91.2 m². Laid out as a square — a small house footprint or a lot — it is about 31.34 ft (31 ft 4 in) on a side.
- 982 is not a sum of two whole-number squares — the prime factor 491 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √982 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 9 × 30 box, because 1² + 9² + 30² = 982.
Square roots near √982 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √979 | √979 | 31.2890 | No |
| √980 | 14√5 | 31.3050 | No |
| √981 | 3√109 | 31.3209 | No |
| √982 | √982 | 31.3369 | No |
| √983 | √983 | 31.3528 | No |
| √984 | 2√246 | 31.3688 | No |
| √985 | √985 | 31.3847 | No |
- The cube root of 982 is about 9.939636.
- Squaring undoes the root: (√982)² = 982, while 982² = 964,324 — the number whose square root is 982.
Frequently asked questions
What is the square root of 982?
The square root of 982 is √982, about 31.3368792320. The negative root, −31.336879, also squares to 982.
Is the square root of 982 rational or irrational?
Irrational. 982 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 √982 be simplified?
No. 982 = 2 × 491 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √982 rounded to two decimal places?
√982 ≈ 31.34 to two decimal places (31.3 to one, 31.337 to three). Check: 31.34² = 982.1956, close to 982.