√255 at a glance
- Exact value
- √255
- Decimal (10 places)
- 15.9687194227
- Rounded
- 16.0 · 15.97 · 15.969
- Perfect square?
- No — between 15² and 16²
- Rational?
- Irrational
- Both square roots
- ±15.968719
- Prime factorization
- 3 × 5 × 17
- Cube root
- 6.341326
How to simplify √255
The prime factorization of 255 is 3 × 5 × 17. Every prime appears only once, so there is no pair to bring outside the radical — √255 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 255, 3, 5 and 17 appear an odd number of times, so √255 is irrational and 15.9687194227 is a rounded value.
Where √255 sits between perfect squares
225 = 15² and 256 = 16² are the nearest perfect squares, so √255 lies between 15 and 16. 255 is 30 above 225 and 1 below 256, so the root is closer to 16.
- Straight line between 225 and 256: 15.9677 (0.01% low)
- Tangent from 15, i.e. 15 + 30 ÷ 30: 16.0000 (0.2% high)
- Tangent from 16, i.e. 16 − 1 ÷ 32: 15.9688 (0% high)
For √255 the tangent at 16 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 255 is just 1 below 256.
Finding √255 with the Babylonian method
If a guess is too big, 255 ÷ 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√255) in one step.
Start from the nearest whole number, 16 (16² = 256):
| Step | Guess x | 255 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 16.0000000000 | 15.9375000000 | 15.9687500000 | 4 |
| 2 | 15.9687500000 | 15.9686888454 | 15.9687194227 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √255 = 15.9687194227 to every decimal shown.
√255 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √255 the pattern is [15; 1, 30] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √255 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 | 9.7 × 10⁻¹ |
| 16/1 | 16.0000000000 | 3.1 × 10⁻² |
| 495/31 | 15.9677419355 | 9.8 × 10⁻⁴ |
| 511/32 | 15.9687500000 | 3.1 × 10⁻⁵ |
| 15,825/991 | 15.9687184662 | 9.6 × 10⁻⁷ |
| 16,336/1,023 | 15.9687194526 | 3.0 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 255y² = 1. Its smallest solution in positive whole numbers is x = 16, y = 1.
√255 in geometry and everyday measurements
- A square patio or deck of 255 square feet is about 15.97 ft (16 ft) on each side, so edging all the way around takes 4 × √255 ≈ 63.9 ft.
- 255 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √255 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √255 as its space diagonal.
Square roots near √255 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √252 | 6√7 | 15.8745 | No |
| √253 | √253 | 15.9060 | No |
| √254 | √254 | 15.9374 | No |
| √255 | √255 | 15.9687 | No |
| √256 | 16 | 16.0000 | Yes |
| √257 | √257 | 16.0312 | No |
| √258 | √258 | 16.0624 | No |
- The cube root of 255 is about 6.341326.
- Squaring undoes the root: (√255)² = 255, while 255² = 65,025 — the number whose square root is 255.
Frequently asked questions
What is the square root of 255?
The square root of 255 is √255, about 15.9687194227. The negative root, −15.968719, also squares to 255.
Is the square root of 255 rational or irrational?
Irrational. 255 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 √255 be simplified?
No. 255 = 3 × 5 × 17 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √255 rounded to two decimal places?
√255 ≈ 15.97 to two decimal places (16.0 to one, 15.969 to three). Check: 15.97² = 255.0409, close to 255.