√254 at a glance
- Exact value
- √254
- Decimal (10 places)
- 15.9373774505
- Rounded
- 15.9 · 15.94 · 15.937
- Perfect square?
- No — between 15² and 16²
- Rational?
- Irrational
- Both square roots
- ±15.937377
- Prime factorization
- 2 × 127
- Cube root
- 6.333026
How to simplify √254
The prime factorization of 254 is 2 × 127. Every prime appears only once, so there is no pair to bring outside the radical — √254 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 254, 2 and 127 appear an odd number of times, so √254 is irrational and 15.9373774505 is a rounded value.
Where √254 sits between perfect squares
225 = 15² and 256 = 16² are the nearest perfect squares, so √254 lies between 15 and 16. 254 is 29 above 225 and 2 below 256, so the root is closer to 16.
- Straight line between 225 and 256: 15.9355 (0.01% low)
- Tangent from 15, i.e. 15 + 29 ÷ 30: 15.9667 (0.18% high)
- Tangent from 16, i.e. 16 − 2 ÷ 32: 15.9375 (0% high)
For √254 the tangent at 16 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 254 is just 2 below 256.
Finding √254 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, 16 (16² = 256):
| Step | Guess x | 254 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 16.0000000000 | 15.8750000000 | 15.9375000000 | 3 |
| 2 | 15.9375000000 | 15.9372549020 | 15.9373774510 | 9 |
| 3 | 15.9373774510 | 15.9373774500 | 15.9373774505 | all 10 shown |
The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √254 = 15.9373774505 to every decimal shown.
√254 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √254 the pattern is [15; 1, 14, 1, 30] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √254 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 | 9.4 × 10⁻¹ |
| 16/1 | 16.0000000000 | 6.3 × 10⁻² |
| 239/15 | 15.9333333333 | 4.0 × 10⁻³ |
| 255/16 | 15.9375000000 | 1.2 × 10⁻⁴ |
| 7,889/495 | 15.9373737374 | 3.7 × 10⁻⁶ |
| 8,144/511 | 15.9373776908 | 2.4 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 254y² = 1. Its smallest solution in positive whole numbers is x = 255, y = 16.
√254 in geometry and everyday measurements
- A square patio or deck of 254 square feet is about 15.94 ft (15 ft 11 in) on each side, so edging all the way around takes 4 × √254 ≈ 63.7 ft.
- 254 is not a sum of two whole-number squares — the prime factor 127 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √254 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 5 × 15 box, because 2² + 5² + 15² = 254.
Square roots near √254 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √251 | √251 | 15.8430 | No |
| √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 |
- The cube root of 254 is about 6.333026.
- Squaring undoes the root: (√254)² = 254, while 254² = 64,516 — the number whose square root is 254.
Frequently asked questions
What is the square root of 254?
The square root of 254 is √254, about 15.9373774505. The negative root, −15.937377, also squares to 254.
Is the square root of 254 rational or irrational?
Irrational. 254 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 √254 be simplified?
No. 254 = 2 × 127 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √254 rounded to two decimal places?
√254 ≈ 15.94 to two decimal places (15.9 to one, 15.937 to three). Check: 15.94² = 254.0836, close to 254.