√674 at a glance
- Exact value
- √674
- Decimal (10 places)
- 25.9615099715
- Rounded
- 26.0 · 25.96 · 25.962
- Perfect square?
- No — between 25² and 26²
- Rational?
- Irrational
- Both square roots
- ±25.961510
- Prime factorization
- 2 × 337
- Cube root
- 8.767719
How to simplify √674
The prime factorization of 674 is 2 × 337. Every prime appears only once, so there is no pair to bring outside the radical — √674 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 674, 2 and 337 appear an odd number of times, so √674 is irrational and 25.9615099715 is a rounded value.
Where √674 sits between perfect squares
625 = 25² and 676 = 26² are the nearest perfect squares, so √674 lies between 25 and 26. 674 is 49 above 625 and 2 below 676, so the root is closer to 26.
- Straight line between 625 and 676: 25.9608 (0% low)
- Tangent from 25, i.e. 25 + 49 ÷ 50: 25.9800 (0.07% high)
- Tangent from 26, i.e. 26 − 2 ÷ 52: 25.9615 (0% high)
For √674 the tangent at 26 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 674 is just 2 below 676.
Finding √674 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 | 674 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 26.0000000000 | 25.9230769231 | 25.9615384615 | 4 |
| 2 | 25.9615384615 | 25.9614814815 | 25.9615099715 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √674 = 25.9615099715 to every decimal shown.
√674 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √674 the pattern is [25; 1, 24, 1, 50] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √674 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 | 9.6 × 10⁻¹ |
| 26/1 | 26.0000000000 | 3.8 × 10⁻² |
| 649/25 | 25.9600000000 | 1.5 × 10⁻³ |
| 675/26 | 25.9615384615 | 2.8 × 10⁻⁵ |
| 34,399/1,325 | 25.9615094340 | 5.4 × 10⁻⁷ |
| 35,074/1,351 | 25.9615099926 | 2.1 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 674y² = 1. Its smallest solution in positive whole numbers is x = 675, y = 26.
√674 in geometry and everyday measurements
- 674 square feet is 62.6 m². Laid out as a square — a small house footprint or a lot — it is about 25.96 ft (26 ft) on a side.
- 674 = 7² + 25², so by the Pythagorean theorem √674 is the diagonal of a 7 × 25 rectangle — and the distance between the points (0, 0) and (7, 25) on a grid.
Square roots near √674 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √671 | √671 | 25.9037 | No |
| √672 | 4√42 | 25.9230 | No |
| √673 | √673 | 25.9422 | No |
| √674 | √674 | 25.9615 | No |
| √675 | 15√3 | 25.9808 | No |
| √676 | 26 | 26.0000 | Yes |
| √677 | √677 | 26.0192 | No |
- The cube root of 674 is about 8.767719.
- Squaring undoes the root: (√674)² = 674, while 674² = 454,276 — the number whose square root is 674.
Frequently asked questions
What is the square root of 674?
The square root of 674 is √674, about 25.9615099715. The negative root, −25.961510, also squares to 674.
Is the square root of 674 rational or irrational?
Irrational. 674 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 √674 be simplified?
No. 674 = 2 × 337 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √674 rounded to two decimal places?
√674 ≈ 25.96 to two decimal places (26.0 to one, 25.962 to three). Check: 25.96² = 673.9216, close to 674.