√611 at a glance
- Exact value
- √611
- Decimal (10 places)
- 24.7184141886
- Rounded
- 24.7 · 24.72 · 24.718
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.718414
- Prime factorization
- 13 × 47
- Cube root
- 8.485558
How to simplify √611
The prime factorization of 611 is 13 × 47. Every prime appears only once, so there is no pair to bring outside the radical — √611 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 611, 13 and 47 appear an odd number of times, so √611 is irrational and 24.7184141886 is a rounded value.
Where √611 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √611 lies between 24 and 25. 611 is 35 above 576 and 14 below 625, so the root is closer to 25.
- Straight line between 576 and 625: 24.7143 (0.02% low)
- Tangent from 24, i.e. 24 + 35 ÷ 48: 24.7292 (0.04% high)
- Tangent from 25, i.e. 25 − 14 ÷ 50: 24.7200 (0.01% high)
For √611 the tangent at 25 wins, missing by only 0.0016. Tangent estimates shine when the number sits close to a perfect square — here 611 is just 14 below 625.
Finding √611 with the Babylonian method
If a guess is too big, 611 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√611) in one step.
Start from the nearest whole number, 25 (25² = 625):
| Step | Guess x | 611 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 25.0000000000 | 24.4400000000 | 24.7200000000 | 2 |
| 2 | 24.7200000000 | 24.7168284790 | 24.7184142395 | 7 |
| 3 | 24.7184142395 | 24.7184141378 | 24.7184141886 | 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 √611 = 24.7184141886 to every decimal shown.
√611 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √611 the pattern is [24; 1, 2, 1, 1, 4, 2, 1, 2, 4, 1, 1, 2, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √611 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 | 7.2 × 10⁻¹ |
| 25/1 | 25.0000000000 | 2.8 × 10⁻¹ |
| 74/3 | 24.6666666667 | 5.2 × 10⁻² |
| 99/4 | 24.7500000000 | 3.2 × 10⁻² |
| 173/7 | 24.7142857143 | 4.1 × 10⁻³ |
| 791/32 | 24.7187500000 | 3.4 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 611y² = 1. Its smallest solution in positive whole numbers is x = 236,926, y = 9,585.
√611 in geometry and everyday measurements
- A square garage floor of 611 square feet measures about 24.72 ft (24 ft 9 in) per side, and its corner-to-corner diagonal is √1222 ≈ 35 ft.
- 611 is not a sum of two whole-number squares — the prime factor 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √611 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 9 × 23 box, because 1² + 9² + 23² = 611.
Square roots near √611 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √613 | √613 | 24.7588 | No |
| √614 | √614 | 24.7790 | No |
- The cube root of 611 is about 8.485558.
- Squaring undoes the root: (√611)² = 611, while 611² = 373,321 — the number whose square root is 611.
Frequently asked questions
What is the square root of 611?
The square root of 611 is √611, about 24.7184141886. The negative root, −24.718414, also squares to 611.
Is the square root of 611 rational or irrational?
Irrational. 611 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 √611 be simplified?
No. 611 = 13 × 47 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √611 rounded to two decimal places?
√611 ≈ 24.72 to two decimal places (24.7 to one, 24.718 to three). Check: 24.72² = 611.0784, close to 611.