√633 at a glance
- Exact value
- √633
- Decimal (10 places)
- 25.1594912508
- Rounded
- 25.2 · 25.16 · 25.159
- Perfect square?
- No — between 25² and 26²
- Rational?
- Irrational
- Both square roots
- ±25.159491
- Prime factorization
- 3 × 211
- Cube root
- 8.586205
How to simplify √633
The prime factorization of 633 is 3 × 211. Every prime appears only once, so there is no pair to bring outside the radical — √633 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 633, 3 and 211 appear an odd number of times, so √633 is irrational and 25.1594912508 is a rounded value.
Where √633 sits between perfect squares
625 = 25² and 676 = 26² are the nearest perfect squares, so √633 lies between 25 and 26. 633 is 8 above 625 and 43 below 676, so the root is closer to 25.
- Straight line between 625 and 676: 25.1569 (0.01% low)
- Tangent from 25, i.e. 25 + 8 ÷ 50: 25.1600 (0% high)
- Tangent from 26, i.e. 26 − 43 ÷ 52: 25.1731 (0.05% high)
For √633 the tangent at 25 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 633 is just 8 above 625.
Finding √633 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 633: following the tangent line down to zero simplifies to averaging x with 633 ÷ x.
Start from the nearest whole number, 25 (25² = 625):
| Step | Guess x | 633 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 25.0000000000 | 25.3200000000 | 25.1600000000 | 3 |
| 2 | 25.1600000000 | 25.1589825119 | 25.1594912560 | 8 |
| 3 | 25.1594912560 | 25.1594912457 | 25.1594912508 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √633 = 25.1594912508 to every decimal shown.
√633 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √633 the pattern is [25; 6, 3, 1, 2, 2, 1, 1, 2, 16, 2, 1, 1, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √633 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 | 1.6 × 10⁻¹ |
| 151/6 | 25.1666666667 | 7.2 × 10⁻³ |
| 478/19 | 25.1578947368 | 1.6 × 10⁻³ |
| 629/25 | 25.1600000000 | 5.1 × 10⁻⁴ |
| 1,736/69 | 25.1594202899 | 7.1 × 10⁻⁵ |
| 4,101/163 | 25.1595092025 | 1.8 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 633y² = 1. Its smallest solution in positive whole numbers is x = 440,772,247, y = 17,519,124.
√633 in geometry and everyday measurements
- A square garage floor of 633 square feet measures about 25.16 ft (25 ft 2 in) per side, and its corner-to-corner diagonal is √1266 ≈ 35.6 ft.
- 633 is not a sum of two whole-number squares — the prime factor 3 and 211 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √633 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 2 × 25 box, because 2² + 2² + 25² = 633.
Square roots near √633 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √630 | 3√70 | 25.0998 | No |
| √631 | √631 | 25.1197 | No |
| √632 | 2√158 | 25.1396 | No |
| √633 | √633 | 25.1595 | No |
| √634 | √634 | 25.1794 | No |
| √635 | √635 | 25.1992 | No |
| √636 | 2√159 | 25.2190 | No |
- The cube root of 633 is about 8.586205.
- Squaring undoes the root: (√633)² = 633, while 633² = 400,689 — the number whose square root is 633.
Frequently asked questions
What is the square root of 633?
The square root of 633 is √633, about 25.1594912508. The negative root, −25.159491, also squares to 633.
Is the square root of 633 rational or irrational?
Irrational. 633 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 √633 be simplified?
No. 633 = 3 × 211 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √633 rounded to two decimal places?
√633 ≈ 25.16 to two decimal places (25.2 to one, 25.159 to three). Check: 25.16² = 633.0256, close to 633.