√609 at a glance
- Exact value
- √609
- Decimal (10 places)
- 24.6779253585
- Rounded
- 24.7 · 24.68 · 24.678
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.677925
- Prime factorization
- 3 × 7 × 29
- Cube root
- 8.476289
How to simplify √609
The prime factorization of 609 is 3 × 7 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √609 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 609, 3, 7 and 29 appear an odd number of times, so √609 is irrational and 24.6779253585 is a rounded value.
Where √609 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √609 lies between 24 and 25. 609 is 33 above 576 and 16 below 625, so the root is closer to 25.
- Straight line between 576 and 625: 24.6735 (0.02% low)
- Tangent from 24, i.e. 24 + 33 ÷ 48: 24.6875 (0.04% high)
- Tangent from 25, i.e. 25 − 16 ÷ 50: 24.6800 (0.01% high)
For √609 the tangent at 25 wins, missing by only 0.0021. Tangent estimates shine when the number sits close to a perfect square — here 609 is just 16 below 625.
Finding √609 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 609: following the tangent line down to zero simplifies to averaging x with 609 ÷ x.
Start from the nearest whole number, 25 (25² = 625):
| Step | Guess x | 609 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 25.0000000000 | 24.3600000000 | 24.6800000000 | 2 |
| 2 | 24.6800000000 | 24.6758508914 | 24.6779254457 | 7 |
| 3 | 24.6779254457 | 24.6779252713 | 24.6779253585 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √609 = 24.6779253585 to every decimal shown.
√609 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √609 the pattern is [24; 1, 2, 9, 1, 1, 6, 1, 1, 9, 2, 1, 48] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √609 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 |
|---|---|---|
| 24/1 | 24.0000000000 | 6.8 × 10⁻¹ |
| 25/1 | 25.0000000000 | 3.2 × 10⁻¹ |
| 74/3 | 24.6666666667 | 1.1 × 10⁻² |
| 691/28 | 24.6785714286 | 6.5 × 10⁻⁴ |
| 765/31 | 24.6774193548 | 5.1 × 10⁻⁴ |
| 1,456/59 | 24.6779661017 | 4.1 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 609y² = 1. Its smallest solution in positive whole numbers is x = 605,695, y = 24,544.
√609 in geometry and everyday measurements
- A square garage floor of 609 square feet measures about 24.68 ft (24 ft 8 in) per side, and its corner-to-corner diagonal is √1218 ≈ 34.9 ft.
- 609 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √609 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 11 × 22 box, because 2² + 11² + 22² = 609.
Square roots near √609 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √606 | √606 | 24.6171 | No |
| √607 | √607 | 24.6374 | No |
| √608 | 4√38 | 24.6577 | No |
| √609 | √609 | 24.6779 | No |
| √610 | √610 | 24.6982 | No |
| √611 | √611 | 24.7184 | No |
| √612 | 6√17 | 24.7386 | No |
- The cube root of 609 is about 8.476289.
- Squaring undoes the root: (√609)² = 609, while 609² = 370,881 — the number whose square root is 609.
Frequently asked questions
What is the square root of 609?
The square root of 609 is √609, about 24.6779253585. The negative root, −24.677925, also squares to 609.
Is the square root of 609 rational or irrational?
Irrational. 609 is not a perfect square — it falls between 576 and 625 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √609 be simplified?
No. 609 = 3 × 7 × 29 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √609 rounded to two decimal places?
√609 ≈ 24.68 to two decimal places (24.7 to one, 24.678 to three). Check: 24.68² = 609.1024, close to 609.