√678 at a glance
- Exact value
- √678
- Decimal (10 places)
- 26.0384331326
- Rounded
- 26.0 · 26.04 · 26.038
- Perfect square?
- No — between 26² and 27²
- Rational?
- Irrational
- Both square roots
- ±26.038433
- Prime factorization
- 2 × 3 × 113
- Cube root
- 8.785030
How to simplify √678
The prime factorization of 678 is 2 × 3 × 113. Every prime appears only once, so there is no pair to bring outside the radical — √678 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 678, 2, 3 and 113 appear an odd number of times, so √678 is irrational and 26.0384331326 is a rounded value.
Where √678 sits between perfect squares
676 = 26² and 729 = 27² are the nearest perfect squares, so √678 lies between 26 and 27. 678 is 2 above 676 and 51 below 729, so the root is closer to 26.
- Straight line between 676 and 729: 26.0377 (0% low)
- Tangent from 26, i.e. 26 + 2 ÷ 52: 26.0385 (0% high)
- Tangent from 27, i.e. 27 − 51 ÷ 54: 26.0556 (0.07% high)
For √678 the tangent at 26 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 678 is just 2 above 676.
Finding √678 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 | 678 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 26.0000000000 | 26.0769230769 | 26.0384615385 | 4 |
| 2 | 26.0384615385 | 26.0384047267 | 26.0384331326 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √678 = 26.0384331326 to every decimal shown.
√678 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √678 the pattern is [26; 26, 52] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √678 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 | 3.8 × 10⁻² |
| 677/26 | 26.0384615385 | 2.8 × 10⁻⁵ |
| 35,230/1,353 | 26.0384331116 | 2.1 × 10⁻⁸ |
| 916,657/35,204 | 26.0384331326 | < 10⁻¹⁰ |
| 47,701,394/1,831,961 | 26.0384331326 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 678y² = 1. Its smallest solution in positive whole numbers is x = 677, y = 26.
√678 in geometry and everyday measurements
- 678 square feet is 63 m². Laid out as a square — a small house footprint or a lot — it is about 26.04 ft (26 ft) on a side.
- 678 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 √678 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 26 box, because 1² + 1² + 26² = 678.
Square roots near √678 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √675 | 15√3 | 25.9808 | No |
| √676 | 26 | 26.0000 | Yes |
| √677 | √677 | 26.0192 | No |
| √678 | √678 | 26.0384 | No |
| √679 | √679 | 26.0576 | No |
| √680 | 2√170 | 26.0768 | No |
| √681 | √681 | 26.0960 | No |
- The cube root of 678 is about 8.785030.
- Squaring undoes the root: (√678)² = 678, while 678² = 459,684 — the number whose square root is 678.
Frequently asked questions
What is the square root of 678?
The square root of 678 is √678, about 26.0384331326. The negative root, −26.038433, also squares to 678.
Is the square root of 678 rational or irrational?
Irrational. 678 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 √678 be simplified?
No. 678 = 2 × 3 × 113 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √678 rounded to two decimal places?
√678 ≈ 26.04 to two decimal places (26.0 to one, 26.038 to three). Check: 26.04² = 678.0816, close to 678.