√665 at a glance
- Exact value
- √665
- Decimal (10 places)
- 25.7875939165
- Rounded
- 25.8 · 25.79 · 25.788
- Perfect square?
- No — between 25² and 26²
- Rational?
- Irrational
- Both square roots
- ±25.787594
- Prime factorization
- 5 × 7 × 19
- Cube root
- 8.728519
How to simplify √665
The prime factorization of 665 is 5 × 7 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √665 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 665, 5, 7 and 19 appear an odd number of times, so √665 is irrational and 25.7875939165 is a rounded value.
Where √665 sits between perfect squares
625 = 25² and 676 = 26² are the nearest perfect squares, so √665 lies between 25 and 26. 665 is 40 above 625 and 11 below 676, so the root is closer to 26.
- Straight line between 625 and 676: 25.7843 (0.01% low)
- Tangent from 25, i.e. 25 + 40 ÷ 50: 25.8000 (0.05% high)
- Tangent from 26, i.e. 26 − 11 ÷ 52: 25.7885 (0% high)
For √665 the tangent at 26 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 665 is just 11 below 676.
Finding √665 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 665: following the tangent line down to zero simplifies to averaging x with 665 ÷ x.
Start from the nearest whole number, 26 (26² = 676):
| Step | Guess x | 665 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 26.0000000000 | 25.5769230769 | 25.7884615385 | 3 |
| 2 | 25.7884615385 | 25.7867263236 | 25.7875939311 | 7 |
| 3 | 25.7875939311 | 25.7875939019 | 25.7875939165 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √665 = 25.7875939165 to every decimal shown.
√665 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √665 the pattern is [25; 1, 3, 1, 2, 2, 2, 1, 3, 1, 50] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √665 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 | 7.9 × 10⁻¹ |
| 26/1 | 26.0000000000 | 2.1 × 10⁻¹ |
| 103/4 | 25.7500000000 | 3.8 × 10⁻² |
| 129/5 | 25.8000000000 | 1.2 × 10⁻² |
| 361/14 | 25.7857142857 | 1.9 × 10⁻³ |
| 851/33 | 25.7878787879 | 2.8 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 665y² = 1. Its smallest solution in positive whole numbers is x = 13,719, y = 532.
√665 in geometry and everyday measurements
- 665 square feet is 61.8 m². Laid out as a square — a small house footprint or a lot — it is about 25.79 ft (25 ft 9 in) on a side.
- 665 is not a sum of two whole-number squares — the prime factor 7 and 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √665 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 6 × 25 box, because 2² + 6² + 25² = 665.
Square roots near √665 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √662 | √662 | 25.7294 | No |
| √663 | √663 | 25.7488 | No |
| √664 | 2√166 | 25.7682 | No |
| √665 | √665 | 25.7876 | No |
| √666 | 3√74 | 25.8070 | No |
| √667 | √667 | 25.8263 | No |
| √668 | 2√167 | 25.8457 | No |
- The cube root of 665 is about 8.728519.
- Squaring undoes the root: (√665)² = 665, while 665² = 442,225 — the number whose square root is 665.
Frequently asked questions
What is the square root of 665?
The square root of 665 is √665, about 25.7875939165. The negative root, −25.787594, also squares to 665.
Is the square root of 665 rational or irrational?
Irrational. 665 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 √665 be simplified?
No. 665 = 5 × 7 × 19 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √665 rounded to two decimal places?
√665 ≈ 25.79 to two decimal places (25.8 to one, 25.788 to three). Check: 25.79² = 665.1241, close to 665.