√686 at a glance
- Exact value
- 7√14
- Decimal (10 places)
- 26.1916017074
- Rounded
- 26.2 · 26.19 · 26.192
- Perfect square?
- No — between 26² and 27²
- Rational?
- Irrational
- Both square roots
- ±26.191602
- Prime factorization
- 2 × 7³
- Cube root
- 8.819447
How to simplify √686
Look for the largest perfect square that divides 686. Here it is 49 (7²), because 686 = 49 × 14 and 14 has no square factor left:
The prime factorization tells the same story: 686 = 2 × 7³. Each pair of equal primes leaves the radical as one factor, so 7 comes out and 2 × 7 stays inside.
Check: (7√14)² = 7² × 14 = 49 × 14 = 686. As a decimal, 7√14 = 7 × 3.7416573868 ≈ 26.1916017074.
Where √686 sits between perfect squares
676 = 26² and 729 = 27² are the nearest perfect squares, so √686 lies between 26 and 27. 686 is 10 above 676 and 43 below 729, so the root is closer to 26.
- Straight line between 676 and 729: 26.1887 (0.01% low)
- Tangent from 26, i.e. 26 + 10 ÷ 52: 26.1923 (0% high)
- Tangent from 27, i.e. 27 − 43 ÷ 54: 26.2037 (0.05% high)
For √686 the tangent at 26 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 686 is just 10 above 676.
Finding √686 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, 26 (26² = 676):
| Step | Guess x | 686 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 26.0000000000 | 26.3846153846 | 26.1923076923 | 3 |
| 2 | 26.1923076923 | 26.1908957416 | 26.1916017169 | 8 |
| 3 | 26.1916017169 | 26.1916016979 | 26.1916017074 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √686 = 26.1916017074 to every decimal shown.
√686 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √686 the pattern is [26; 5, 4, 1, 1, 3, 2, 10, 26, 10, 2, 3, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √686 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 |
|---|---|---|
| 26/1 | 26.0000000000 | 1.9 × 10⁻¹ |
| 131/5 | 26.2000000000 | 8.4 × 10⁻³ |
| 550/21 | 26.1904761905 | 1.1 × 10⁻³ |
| 681/26 | 26.1923076923 | 7.1 × 10⁻⁴ |
| 1,231/47 | 26.1914893617 | 1.1 × 10⁻⁴ |
| 4,374/167 | 26.1916167665 | 1.5 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 686y² = 1. Its smallest solution in positive whole numbers is x = 10,850,138,895, y = 414,260,228.
√686 in geometry and everyday measurements
- 686 square feet is 63.7 m². Laid out as a square — a small house footprint or a lot — it is about 26.19 ft (26 ft 2 in) on a side.
- 686 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √686 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 26 box, because 1² + 3² + 26² = 686.
- Since √686 = 7√14, a length of √686 is exactly 7 copies of the length √14 laid end to end.
Square roots near √686 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √683 | √683 | 26.1343 | No |
| √684 | 6√19 | 26.1534 | No |
| √685 | √685 | 26.1725 | No |
| √686 | 7√14 | 26.1916 | No |
| √687 | √687 | 26.2107 | No |
| √688 | 4√43 | 26.2298 | No |
| √689 | √689 | 26.2488 | No |
- The cube root of 686 is about 8.819447.
- Squaring undoes the root: (√686)² = 686, while 686² = 470,596 — the number whose square root is 686.
Frequently asked questions
What is the square root of 686?
The square root of 686 is 7√14 in simplest radical form, which is about 26.1916017074. The negative root, −26.191602, also squares to 686.
Is the square root of 686 rational or irrational?
Irrational. 686 is not a perfect square — it falls between 676 and 729 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √686 be simplified?
Yes. The largest perfect square dividing 686 is 49, so √686 = √49 × √14 = 7√14.
What is √686 rounded to two decimal places?
√686 ≈ 26.19 to two decimal places (26.2 to one, 26.192 to three). Check: 26.19² = 685.9161, close to 686.