√235 at a glance
- Exact value
- √235
- Decimal (10 places)
- 15.3297097168
- Rounded
- 15.3 · 15.33 · 15.330
- Perfect square?
- No — between 15² and 16²
- Rational?
- Irrational
- Both square roots
- ±15.329710
- Prime factorization
- 5 × 47
- Cube root
- 6.171006
How to simplify √235
The prime factorization of 235 is 5 × 47. Every prime appears only once, so there is no pair to bring outside the radical — √235 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 235, 5 and 47 appear an odd number of times, so √235 is irrational and 15.3297097168 is a rounded value.
Where √235 sits between perfect squares
225 = 15² and 256 = 16² are the nearest perfect squares, so √235 lies between 15 and 16. 235 is 10 above 225 and 21 below 256, so the root is closer to 15.
- Straight line between 225 and 256: 15.3226 (0.05% low)
- Tangent from 15, i.e. 15 + 10 ÷ 30: 15.3333 (0.02% high)
- Tangent from 16, i.e. 16 − 21 ÷ 32: 15.3438 (0.09% high)
For √235 the tangent at 15 wins, missing by only 0.0036. Tangent estimates shine when the number sits close to a perfect square — here 235 is just 10 above 225.
Finding √235 with the Babylonian method
If a guess is too big, 235 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√235) in one step.
Start from the nearest whole number, 15 (15² = 225):
| Step | Guess x | 235 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 15.0000000000 | 15.6666666667 | 15.3333333333 | 2 |
| 2 | 15.3333333333 | 15.3260869565 | 15.3297101449 | 6 |
| 3 | 15.3297101449 | 15.3297092886 | 15.3297097168 | 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 √235 = 15.3297097168 to every decimal shown.
√235 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √235 the pattern is [15; 3, 30] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √235 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 | 3.3 × 10⁻¹ |
| 46/3 | 15.3333333333 | 3.6 × 10⁻³ |
| 1,395/91 | 15.3296703297 | 3.9 × 10⁻⁵ |
| 4,231/276 | 15.3297101449 | 4.3 × 10⁻⁷ |
| 128,325/8,371 | 15.3297097121 | 4.7 × 10⁻⁹ |
| 389,206/25,389 | 15.3297097168 | 5.1 × 10⁻¹¹ |
The same fractions solve Pell’s equation, x² − 235y² = 1. Its smallest solution in positive whole numbers is x = 46, y = 3.
√235 in geometry and everyday measurements
- A square patio or deck of 235 square feet is about 15.33 ft (15 ft 4 in) on each side, so edging all the way around takes 4 × √235 ≈ 61.3 ft.
- 235 is not a sum of two whole-number squares — the prime factor 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √235 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 15 box, because 1² + 3² + 15² = 235.
Square roots near √235 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √232 | 2√58 | 15.2315 | No |
| √233 | √233 | 15.2643 | No |
| √234 | 3√26 | 15.2971 | No |
| √235 | √235 | 15.3297 | No |
| √236 | 2√59 | 15.3623 | No |
| √237 | √237 | 15.3948 | No |
| √238 | √238 | 15.4272 | No |
- The cube root of 235 is about 6.171006.
- Four times the radicand doubles the root: √940 = 2 × √235 ≈ 30.659419.
Frequently asked questions
What is the square root of 235?
The square root of 235 is √235, about 15.3297097168. The negative root, −15.329710, also squares to 235.
Is the square root of 235 rational or irrational?
Irrational. 235 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 √235 be simplified?
No. 235 = 5 × 47 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √235 rounded to two decimal places?
√235 ≈ 15.33 to two decimal places (15.3 to one, 15.330 to three). Check: 15.33² = 235.0089, close to 235.