√627 at a glance
- Exact value
- √627
- Decimal (10 places)
- 25.0399680511
- Rounded
- 25.0 · 25.04 · 25.040
- Perfect square?
- No — between 25² and 26²
- Rational?
- Irrational
- Both square roots
- ±25.039968
- Prime factorization
- 3 × 11 × 19
- Cube root
- 8.558990
How to simplify √627
The prime factorization of 627 is 3 × 11 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √627 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 627, 3, 11 and 19 appear an odd number of times, so √627 is irrational and 25.0399680511 is a rounded value.
Where √627 sits between perfect squares
625 = 25² and 676 = 26² are the nearest perfect squares, so √627 lies between 25 and 26. 627 is 2 above 625 and 49 below 676, so the root is closer to 25.
- Straight line between 625 and 676: 25.0392 (0% low)
- Tangent from 25, i.e. 25 + 2 ÷ 50: 25.0400 (0% high)
- Tangent from 26, i.e. 26 − 49 ÷ 52: 25.0577 (0.07% high)
For √627 the tangent at 25 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 627 is just 2 above 625.
Finding √627 with the Babylonian method
If a guess is too big, 627 ÷ 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√627) in one step.
Start from the nearest whole number, 25 (25² = 625):
| Step | Guess x | 627 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 25.0000000000 | 25.0800000000 | 25.0400000000 | 4 |
| 2 | 25.0400000000 | 25.0399361022 | 25.0399680511 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √627 = 25.0399680511 to every decimal shown.
√627 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √627 the pattern is [25; 25, 50] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √627 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 | 4.0 × 10⁻² |
| 626/25 | 25.0400000000 | 3.2 × 10⁻⁵ |
| 31,325/1,251 | 25.0399680256 | 2.6 × 10⁻⁸ |
| 783,751/31,300 | 25.0399680511 | < 10⁻¹⁰ |
| 39,218,875/1,566,251 | 25.0399680511 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 627y² = 1. Its smallest solution in positive whole numbers is x = 626, y = 25.
√627 in geometry and everyday measurements
- A square garage floor of 627 square feet measures about 25.04 ft (25 ft) per side, and its corner-to-corner diagonal is √1254 ≈ 35.4 ft.
- 627 is not a sum of two whole-number squares — the prime factor 3, 11 and 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √627 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 25 box, because 1² + 1² + 25² = 627.
Square roots near √627 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √624 | 4√39 | 24.9800 | No |
| √625 | 25 | 25.0000 | Yes |
| √626 | √626 | 25.0200 | No |
| √627 | √627 | 25.0400 | No |
| √628 | 2√157 | 25.0599 | No |
| √629 | √629 | 25.0799 | No |
| √630 | 3√70 | 25.0998 | No |
- The cube root of 627 is about 8.558990.
- Squaring undoes the root: (√627)² = 627, while 627² = 393,129 — the number whose square root is 627.
Frequently asked questions
What is the square root of 627?
The square root of 627 is √627, about 25.0399680511. The negative root, −25.039968, also squares to 627.
Is the square root of 627 rational or irrational?
Irrational. 627 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 √627 be simplified?
No. 627 = 3 × 11 × 19 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √627 rounded to two decimal places?
√627 ≈ 25.04 to two decimal places (25.0 to one, 25.040 to three). Check: 25.04² = 627.0016, close to 627.