√671 at a glance
- Exact value
- √671
- Decimal (10 places)
- 25.9036676940
- Rounded
- 25.9 · 25.90 · 25.904
- Perfect square?
- No — between 25² and 26²
- Rational?
- Irrational
- Both square roots
- ±25.903668
- Prime factorization
- 11 × 61
- Cube root
- 8.754691
How to simplify √671
The prime factorization of 671 is 11 × 61. Every prime appears only once, so there is no pair to bring outside the radical — √671 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 671, 11 and 61 appear an odd number of times, so √671 is irrational and 25.9036676940 is a rounded value.
Where √671 sits between perfect squares
625 = 25² and 676 = 26² are the nearest perfect squares, so √671 lies between 25 and 26. 671 is 46 above 625 and 5 below 676, so the root is closer to 26.
- Straight line between 625 and 676: 25.9020 (0.01% low)
- Tangent from 25, i.e. 25 + 46 ÷ 50: 25.9200 (0.06% high)
- Tangent from 26, i.e. 26 − 5 ÷ 52: 25.9038 (0% high)
For √671 the tangent at 26 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 671 is just 5 below 676.
Finding √671 with the Babylonian method
If a guess is too big, 671 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√671) in one step.
Start from the nearest whole number, 26 (26² = 676):
| Step | Guess x | 671 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 26.0000000000 | 25.8076923077 | 25.9038461538 | 3 |
| 2 | 25.9038461538 | 25.9034892353 | 25.9036676946 | 9 |
| 3 | 25.9036676946 | 25.9036676934 | 25.9036676940 | 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 √671 = 25.9036676940 to every decimal shown.
√671 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √671 the pattern is [25; 1, 9, 2, 1, 1, 1, 2, 9, 1, 50] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √671 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 | 9.0 × 10⁻¹ |
| 26/1 | 26.0000000000 | 9.6 × 10⁻² |
| 259/10 | 25.9000000000 | 3.7 × 10⁻³ |
| 544/21 | 25.9047619048 | 1.1 × 10⁻³ |
| 803/31 | 25.9032258065 | 4.4 × 10⁻⁴ |
| 1,347/52 | 25.9038461538 | 1.8 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 671y² = 1. Its smallest solution in positive whole numbers is x = 58,620, y = 2,263.
√671 in geometry and everyday measurements
- 671 square feet is 62.3 m². Laid out as a square — a small house footprint or a lot — it is about 25.9 ft (25 ft 11 in) on a side.
- 671 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √671 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √671 as its space diagonal.
Square roots near √671 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √668 | 2√167 | 25.8457 | No |
| √669 | √669 | 25.8650 | No |
| √670 | √670 | 25.8844 | No |
| √671 | √671 | 25.9037 | No |
| √672 | 4√42 | 25.9230 | No |
| √673 | √673 | 25.9422 | No |
| √674 | √674 | 25.9615 | No |
- The cube root of 671 is about 8.754691.
- Squaring undoes the root: (√671)² = 671, while 671² = 450,241 — the number whose square root is 671.
Frequently asked questions
What is the square root of 671?
The square root of 671 is √671, about 25.9036676940. The negative root, −25.903668, also squares to 671.
Is the square root of 671 rational or irrational?
Irrational. 671 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 √671 be simplified?
No. 671 = 11 × 61 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √671 rounded to two decimal places?
√671 ≈ 25.90 to two decimal places (25.9 to one, 25.904 to three). Check: 25.90² = 670.81, close to 671.