√646 at a glance
- Exact value
- √646
- Decimal (10 places)
- 25.4165300543
- Rounded
- 25.4 · 25.42 · 25.417
- Perfect square?
- No — between 25² and 26²
- Rational?
- Irrational
- Both square roots
- ±25.416530
- Prime factorization
- 2 × 17 × 19
- Cube root
- 8.644585
How to simplify √646
The prime factorization of 646 is 2 × 17 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √646 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 646, 2, 17 and 19 appear an odd number of times, so √646 is irrational and 25.4165300543 is a rounded value.
Where √646 sits between perfect squares
625 = 25² and 676 = 26² are the nearest perfect squares, so √646 lies between 25 and 26. 646 is 21 above 625 and 30 below 676, so the root is closer to 25.
- Straight line between 625 and 676: 25.4118 (0.02% low)
- Tangent from 25, i.e. 25 + 21 ÷ 50: 25.4200 (0.01% high)
- Tangent from 26, i.e. 26 − 30 ÷ 52: 25.4231 (0.03% high)
For √646 the tangent at 25 wins, missing by only 0.0035. Tangent estimates shine when the number sits close to a perfect square — here 646 is just 21 above 625.
Finding √646 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, 25 (25² = 625):
| Step | Guess x | 646 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 25.0000000000 | 25.8400000000 | 25.4200000000 | 2 |
| 2 | 25.4200000000 | 25.4130605822 | 25.4165302911 | 6 |
| 3 | 25.4165302911 | 25.4165298174 | 25.4165300543 | 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 √646 = 25.4165300543 to every decimal shown.
√646 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √646 the pattern is [25; 2, 2, 2, 50] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √646 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 |
|---|---|---|
| 25/1 | 25.0000000000 | 4.2 × 10⁻¹ |
| 51/2 | 25.5000000000 | 8.3 × 10⁻² |
| 127/5 | 25.4000000000 | 1.7 × 10⁻² |
| 305/12 | 25.4166666667 | 1.4 × 10⁻⁴ |
| 15,377/605 | 25.4165289256 | 1.1 × 10⁻⁶ |
| 31,059/1,222 | 25.4165302782 | 2.2 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 646y² = 1. Its smallest solution in positive whole numbers is x = 305, y = 12.
√646 in geometry and everyday measurements
- A square garage floor of 646 square feet measures about 25.42 ft (25 ft 5 in) per side, and its corner-to-corner diagonal is √1292 ≈ 35.9 ft.
- 646 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √646 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 14 × 21 box, because 3² + 14² + 21² = 646.
Square roots near √646 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √643 | √643 | 25.3574 | No |
| √644 | 2√161 | 25.3772 | No |
| √645 | √645 | 25.3969 | No |
| √646 | √646 | 25.4165 | No |
| √647 | √647 | 25.4362 | No |
| √648 | 18√2 | 25.4558 | No |
| √649 | √649 | 25.4755 | No |
- The cube root of 646 is about 8.644585.
- Squaring undoes the root: (√646)² = 646, while 646² = 417,316 — the number whose square root is 646.
Frequently asked questions
What is the square root of 646?
The square root of 646 is √646, about 25.4165300543. The negative root, −25.416530, also squares to 646.
Is the square root of 646 rational or irrational?
Irrational. 646 is not a perfect square — it falls between 625 and 676 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √646 be simplified?
No. 646 = 2 × 17 × 19 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √646 rounded to two decimal places?
√646 ≈ 25.42 to two decimal places (25.4 to one, 25.417 to three). Check: 25.42² = 646.1764, close to 646.