√630 at a glance
- Exact value
- 3√70
- Decimal (10 places)
- 25.0998007960
- Rounded
- 25.1 · 25.10 · 25.100
- Perfect square?
- No — between 25² and 26²
- Rational?
- Irrational
- Both square roots
- ±25.099801
- Prime factorization
- 2 × 3² × 5 × 7
- Cube root
- 8.572619
How to simplify √630
Look for the largest perfect square that divides 630. Here it is 9 (3²), because 630 = 9 × 70 and 70 has no square factor left:
The prime factorization tells the same story: 630 = 2 × 3² × 5 × 7. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 5 × 7 stays inside.
Check: (3√70)² = 3² × 70 = 9 × 70 = 630. As a decimal, 3√70 = 3 × 8.3666002653 ≈ 25.0998007960.
Where √630 sits between perfect squares
625 = 25² and 676 = 26² are the nearest perfect squares, so √630 lies between 25 and 26. 630 is 5 above 625 and 46 below 676, so the root is closer to 25.
- Straight line between 625 and 676: 25.0980 (0.01% low)
- Tangent from 25, i.e. 25 + 5 ÷ 50: 25.1000 (0% high)
- Tangent from 26, i.e. 26 − 46 ÷ 52: 25.1154 (0.06% high)
For √630 the tangent at 25 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 630 is just 5 above 625.
Finding √630 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, 25 (25² = 625):
| Step | Guess x | 630 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 25.0000000000 | 25.2000000000 | 25.1000000000 | 3 |
| 2 | 25.1000000000 | 25.0996015936 | 25.0998007968 | 9 |
| 3 | 25.0998007968 | 25.0998007952 | 25.0998007960 | 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 √630 = 25.0998007960 to every decimal shown.
√630 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √630 the pattern is [25; 10, 50] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √630 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 | 1.0 × 10⁻¹ |
| 251/10 | 25.1000000000 | 2.0 × 10⁻⁴ |
| 12,575/501 | 25.0998003992 | 4.0 × 10⁻⁷ |
| 126,001/5,020 | 25.0998007968 | 7.9 × 10⁻¹⁰ |
| 6,312,625/251,501 | 25.0998007960 | < 10⁻¹⁰ |
| 63,252,251/2,520,030 | 25.0998007960 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 630y² = 1. Its smallest solution in positive whole numbers is x = 251, y = 10.
√630 in geometry and everyday measurements
- A square garage floor of 630 square feet measures about 25.1 ft (25 ft 1 in) per side, and its corner-to-corner diagonal is √1260 ≈ 35.5 ft.
- 630 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √630 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 25 box, because 1² + 2² + 25² = 630.
- Since √630 = 3√70, a length of √630 is exactly 3 copies of the length √70 laid end to end.
Square roots near √630 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √632 | 2√158 | 25.1396 | No |
| √633 | √633 | 25.1595 | No |
- The cube root of 630 is about 8.572619.
- Squaring undoes the root: (√630)² = 630, while 630² = 396,900 — the number whose square root is 630.
Frequently asked questions
What is the square root of 630?
The square root of 630 is 3√70 in simplest radical form, which is about 25.0998007960. The negative root, −25.099801, also squares to 630.
Is the square root of 630 rational or irrational?
Irrational. 630 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 √630 be simplified?
Yes. The largest perfect square dividing 630 is 9, so √630 = √9 × √70 = 3√70.
What is √630 rounded to two decimal places?
√630 ≈ 25.10 to two decimal places (25.1 to one, 25.100 to three). Check: 25.10² = 630.01, close to 630.