√606 at a glance
- Exact value
- √606
- Decimal (10 places)
- 24.6170672502
- Rounded
- 24.6 · 24.62 · 24.617
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.617067
- Prime factorization
- 2 × 3 × 101
- Cube root
- 8.462348
How to simplify √606
The prime factorization of 606 is 2 × 3 × 101. Every prime appears only once, so there is no pair to bring outside the radical — √606 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 606, 2, 3 and 101 appear an odd number of times, so √606 is irrational and 24.6170672502 is a rounded value.
Where √606 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √606 lies between 24 and 25. 606 is 30 above 576 and 19 below 625, so the root is closer to 25.
- Straight line between 576 and 625: 24.6122 (0.02% low)
- Tangent from 24, i.e. 24 + 30 ÷ 48: 24.6250 (0.03% high)
- Tangent from 25, i.e. 25 − 19 ÷ 50: 24.6200 (0.01% high)
For √606 the tangent at 25 wins, missing by only 0.0029. Tangent estimates shine when the number sits close to a perfect square — here 606 is just 19 below 625.
Finding √606 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 | 606 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 25.0000000000 | 24.2400000000 | 24.6200000000 | 2 |
| 2 | 24.6200000000 | 24.6141348497 | 24.6170674249 | 6 |
| 3 | 24.6170674249 | 24.6170670755 | 24.6170672502 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √606 = 24.6170672502 to every decimal shown.
√606 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √606 the pattern is [24; 1, 1, 1, 1, 1, 1, 2, 1, 9, 8, 9, 1, …] with the block of 20 terms after the semicolon repeating forever (only the first 12 of the 20 are shown). A pattern that never ends is one more proof that √606 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 | 6.2 × 10⁻¹ |
| 25/1 | 25.0000000000 | 3.8 × 10⁻¹ |
| 49/2 | 24.5000000000 | 1.2 × 10⁻¹ |
| 74/3 | 24.6666666667 | 5.0 × 10⁻² |
| 123/5 | 24.6000000000 | 1.7 × 10⁻² |
| 197/8 | 24.6250000000 | 7.9 × 10⁻³ |
The same fractions solve Pell’s equation, x² − 606y² = 1. Its smallest solution in positive whole numbers is x = 42,187,499, y = 1,713,750.
√606 in geometry and everyday measurements
- A square garage floor of 606 square feet measures about 24.62 ft (24 ft 7 in) per side, and its corner-to-corner diagonal is √1212 ≈ 34.8 ft.
- 606 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √606 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 11 × 22 box, because 1² + 11² + 22² = 606.
Square roots near √606 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √603 | 3√67 | 24.5561 | No |
| √604 | 2√151 | 24.5764 | No |
| √605 | 11√5 | 24.5967 | No |
| √606 | √606 | 24.6171 | No |
| √607 | √607 | 24.6374 | No |
| √608 | 4√38 | 24.6577 | No |
| √609 | √609 | 24.6779 | No |
- The cube root of 606 is about 8.462348.
- Squaring undoes the root: (√606)² = 606, while 606² = 367,236 — the number whose square root is 606.
Frequently asked questions
What is the square root of 606?
The square root of 606 is √606, about 24.6170672502. The negative root, −24.617067, also squares to 606.
Is the square root of 606 rational or irrational?
Irrational. 606 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 √606 be simplified?
No. 606 = 2 × 3 × 101 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √606 rounded to two decimal places?
√606 ≈ 24.62 to two decimal places (24.6 to one, 24.617 to three). Check: 24.62² = 606.1444, close to 606.