√628 at a glance
- Exact value
- 2√157
- Decimal (10 places)
- 25.0599281723
- Rounded
- 25.1 · 25.06 · 25.060
- Perfect square?
- No — between 25² and 26²
- Rational?
- Irrational
- Both square roots
- ±25.059928
- Prime factorization
- 2² × 157
- Cube root
- 8.563538
How to simplify √628
Look for the largest perfect square that divides 628. Here it is 4 (2²), because 628 = 4 × 157 and 157 has no square factor left:
The prime factorization tells the same story: 628 = 2² × 157. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 157 stays inside.
Check: (2√157)² = 2² × 157 = 4 × 157 = 628. As a decimal, 2√157 = 2 × 12.5299640861 ≈ 25.0599281723.
Where √628 sits between perfect squares
625 = 25² and 676 = 26² are the nearest perfect squares, so √628 lies between 25 and 26. 628 is 3 above 625 and 48 below 676, so the root is closer to 25.
- Straight line between 625 and 676: 25.0588 (0% low)
- Tangent from 25, i.e. 25 + 3 ÷ 50: 25.0600 (0% high)
- Tangent from 26, i.e. 26 − 48 ÷ 52: 25.0769 (0.07% high)
For √628 the tangent at 25 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 628 is just 3 above 625.
Finding √628 with the Babylonian method
Picture a rectangle with an area of 628 and one side x; the other side must be 628 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √628.
Start from the nearest whole number, 25 (25² = 625):
| Step | Guess x | 628 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 25.0000000000 | 25.1200000000 | 25.0600000000 | 4 |
| 2 | 25.0600000000 | 25.0598563448 | 25.0599281724 | 9 |
| 3 | 25.0599281724 | 25.0599281722 | 25.0599281723 | all 10 shown |
The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √628 = 25.0599281723 to every decimal shown.
√628 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √628 the pattern is [25; 16, 1, 2, 5, 4, 2, 1, 2, 2, 3, 1, 3, …] with the block of 26 terms after the semicolon repeating forever (only the first 12 of the 26 are shown). A pattern that never ends is one more proof that √628 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 | 6.0 × 10⁻² |
| 401/16 | 25.0625000000 | 2.6 × 10⁻³ |
| 426/17 | 25.0588235294 | 1.1 × 10⁻³ |
| 1,253/50 | 25.0600000000 | 7.2 × 10⁻⁵ |
| 6,691/267 | 25.0599250936 | 3.1 × 10⁻⁶ |
| 28,017/1,118 | 25.0599284436 | 2.7 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 628y² = 1. Its smallest solution in positive whole numbers is x = 46,698,728,731,849, y = 1,863,482,146,110 — 14 digits for x, even though 628 is small, which is what makes Pell’s equation famous.
√628 in geometry and everyday measurements
- A square garage floor of 628 square feet measures about 25.06 ft (25 ft 1 in) per side, and its corner-to-corner diagonal is √1256 ≈ 35.4 ft.
- 628 = 12² + 22², so by the Pythagorean theorem √628 is the diagonal of a 12 × 22 rectangle — and the distance between the points (0, 0) and (12, 22) on a grid.
- Since √628 = 2√157, a length of √628 is exactly 2 copies of the length √157 laid end to end.
Square roots near √628 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √631 | √631 | 25.1197 | No |
- The cube root of 628 is about 8.563538.
- Because 628 = 4 × 157, the root is twice √157: 2 × 12.529964 ≈ 25.059928.
Frequently asked questions
What is the square root of 628?
The square root of 628 is 2√157 in simplest radical form, which is about 25.0599281723. The negative root, −25.059928, also squares to 628.
Is the square root of 628 rational or irrational?
Irrational. 628 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 √628 be simplified?
Yes. The largest perfect square dividing 628 is 4, so √628 = √4 × √157 = 2√157.
What is √628 rounded to two decimal places?
√628 ≈ 25.06 to two decimal places (25.1 to one, 25.060 to three). Check: 25.06² = 628.0036, close to 628.