√310 at a glance
- Exact value
- √310
- Decimal (10 places)
- 17.6068168617
- Rounded
- 17.6 · 17.61 · 17.607
- Perfect square?
- No — between 17² and 18²
- Rational?
- Irrational
- Both square roots
- ±17.606817
- Prime factorization
- 2 × 5 × 31
- Cube root
- 6.767899
How to simplify √310
The prime factorization of 310 is 2 × 5 × 31. Every prime appears only once, so there is no pair to bring outside the radical — √310 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 310, 2, 5 and 31 appear an odd number of times, so √310 is irrational and 17.6068168617 is a rounded value.
Where √310 sits between perfect squares
289 = 17² and 324 = 18² are the nearest perfect squares, so √310 lies between 17 and 18. 310 is 21 above 289 and 14 below 324, so the root is closer to 18.
- Straight line between 289 and 324: 17.6000 (0.04% low)
- Tangent from 17, i.e. 17 + 21 ÷ 34: 17.6176 (0.06% high)
- Tangent from 18, i.e. 18 − 14 ÷ 36: 17.6111 (0.02% high)
For √310 the tangent at 18 wins, missing by only 0.0043. Tangent estimates shine when the number sits close to a perfect square — here 310 is just 14 below 324.
Finding √310 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, 18 (18² = 324):
| Step | Guess x | 310 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 18.0000000000 | 17.2222222222 | 17.6111111111 | 2 |
| 2 | 17.6111111111 | 17.6025236593 | 17.6068173852 | 6 |
| 3 | 17.6068173852 | 17.6068163381 | 17.6068168617 | 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 √310 = 17.6068168617 to every decimal shown.
√310 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √310 the pattern is [17; 1, 1, 1, 1, 5, 3, 1, 2, 1, 3, 5, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √310 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 |
|---|---|---|
| 17/1 | 17.0000000000 | 6.1 × 10⁻¹ |
| 18/1 | 18.0000000000 | 3.9 × 10⁻¹ |
| 35/2 | 17.5000000000 | 1.1 × 10⁻¹ |
| 53/3 | 17.6666666667 | 6.0 × 10⁻² |
| 88/5 | 17.6000000000 | 6.8 × 10⁻³ |
| 493/28 | 17.6071428571 | 3.3 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 310y² = 1. Its smallest solution in positive whole numbers is x = 848,719, y = 48,204.
√310 in geometry and everyday measurements
- A square patio or deck of 310 square feet is about 17.61 ft (17 ft 7 in) on each side, so edging all the way around takes 4 × √310 ≈ 70.4 ft.
- 310 is not a sum of two whole-number squares — the prime factor 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √310 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 9 × 15 box, because 2² + 9² + 15² = 310.
Square roots near √310 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √307 | √307 | 17.5214 | No |
| √308 | 2√77 | 17.5499 | No |
| √309 | √309 | 17.5784 | No |
| √310 | √310 | 17.6068 | No |
| √311 | √311 | 17.6352 | No |
| √312 | 2√78 | 17.6635 | No |
| √313 | √313 | 17.6918 | No |
- The cube root of 310 is about 6.767899.
- Squaring undoes the root: (√310)² = 310, while 310² = 96,100 — the number whose square root is 310.
Frequently asked questions
What is the square root of 310?
The square root of 310 is √310, about 17.6068168617. The negative root, −17.606817, also squares to 310.
Is the square root of 310 rational or irrational?
Irrational. 310 is not a perfect square — it falls between 289 and 324 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √310 be simplified?
No. 310 = 2 × 5 × 31 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √310 rounded to two decimal places?
√310 ≈ 17.61 to two decimal places (17.6 to one, 17.607 to three). Check: 17.61² = 310.1121, close to 310.