√230 at a glance
- Exact value
- √230
- Decimal (10 places)
- 15.1657508881
- Rounded
- 15.2 · 15.17 · 15.166
- Perfect square?
- No — between 15² and 16²
- Rational?
- Irrational
- Both square roots
- ±15.165751
- Prime factorization
- 2 × 5 × 23
- Cube root
- 6.126926
How to simplify √230
The prime factorization of 230 is 2 × 5 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √230 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 230, 2, 5 and 23 appear an odd number of times, so √230 is irrational and 15.1657508881 is a rounded value.
Where √230 sits between perfect squares
225 = 15² and 256 = 16² are the nearest perfect squares, so √230 lies between 15 and 16. 230 is 5 above 225 and 26 below 256, so the root is closer to 15.
- Straight line between 225 and 256: 15.1613 (0.03% low)
- Tangent from 15, i.e. 15 + 5 ÷ 30: 15.1667 (0.01% high)
- Tangent from 16, i.e. 16 − 26 ÷ 32: 15.1875 (0.14% high)
For √230 the tangent at 15 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 230 is just 5 above 225.
Finding √230 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, 15 (15² = 225):
| Step | Guess x | 230 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 15.0000000000 | 15.3333333333 | 15.1666666667 | 3 |
| 2 | 15.1666666667 | 15.1648351648 | 15.1657509158 | 7 |
| 3 | 15.1657509158 | 15.1657508605 | 15.1657508881 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √230 = 15.1657508881 to every decimal shown.
√230 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √230 the pattern is [15; 6, 30] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √230 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 |
|---|---|---|
| 15/1 | 15.0000000000 | 1.7 × 10⁻¹ |
| 91/6 | 15.1666666667 | 9.2 × 10⁻⁴ |
| 2,745/181 | 15.1657458564 | 5.0 × 10⁻⁶ |
| 16,561/1,092 | 15.1657509158 | 2.8 × 10⁻⁸ |
| 499,575/32,941 | 15.1657508880 | 1.5 × 10⁻¹⁰ |
| 3,014,011/198,738 | 15.1657508881 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 230y² = 1. Its smallest solution in positive whole numbers is x = 91, y = 6.
√230 in geometry and everyday measurements
- A square patio or deck of 230 square feet is about 15.17 ft (15 ft 2 in) on each side, so edging all the way around takes 4 × √230 ≈ 60.7 ft.
- 230 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √230 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 15 box, because 1² + 2² + 15² = 230.
Square roots near √230 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √227 | √227 | 15.0665 | No |
| √228 | 2√57 | 15.0997 | No |
| √229 | √229 | 15.1327 | No |
| √230 | √230 | 15.1658 | No |
| √231 | √231 | 15.1987 | No |
| √232 | 2√58 | 15.2315 | No |
| √233 | √233 | 15.2643 | No |
- The cube root of 230 is about 6.126926.
- Four times the radicand doubles the root: √920 = 2 × √230 ≈ 30.331502.
Frequently asked questions
What is the square root of 230?
The square root of 230 is √230, about 15.1657508881. The negative root, −15.165751, also squares to 230.
Is the square root of 230 rational or irrational?
Irrational. 230 is not a perfect square — it falls between 225 and 256 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √230 be simplified?
No. 230 = 2 × 5 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √230 rounded to two decimal places?
√230 ≈ 15.17 to two decimal places (15.2 to one, 15.166 to three). Check: 15.17² = 230.1289, close to 230.