√598 at a glance
- Exact value
- √598
- Decimal (10 places)
- 24.4540385213
- Rounded
- 24.5 · 24.45 · 24.454
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.454039
- Prime factorization
- 2 × 13 × 23
- Cube root
- 8.424945
How to simplify √598
The prime factorization of 598 is 2 × 13 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √598 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 598, 2, 13 and 23 appear an odd number of times, so √598 is irrational and 24.4540385213 is a rounded value.
Where √598 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √598 lies between 24 and 25. 598 is 22 above 576 and 27 below 625, so the root is closer to 24.
- Straight line between 576 and 625: 24.4490 (0.02% low)
- Tangent from 24, i.e. 24 + 22 ÷ 48: 24.4583 (0.02% high)
- Tangent from 25, i.e. 25 − 27 ÷ 50: 24.4600 (0.02% high)
For √598 the tangent at 24 wins, missing by only 0.0043. Tangent estimates shine when the number sits close to a perfect square — here 598 is just 22 above 576.
Finding √598 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, 24 (24² = 576):
| Step | Guess x | 598 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 24.0000000000 | 24.9166666667 | 24.4583333333 | 2 |
| 2 | 24.4583333333 | 24.4497444634 | 24.4540388984 | 6 |
| 3 | 24.4540388984 | 24.4540381442 | 24.4540385213 | 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 √598 = 24.4540385213 to every decimal shown.
√598 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √598 the pattern is [24; 2, 4, 1, 15, 2, 15, 1, 4, 2, 48] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √598 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 | 4.5 × 10⁻¹ |
| 49/2 | 24.5000000000 | 4.6 × 10⁻² |
| 220/9 | 24.4444444444 | 9.6 × 10⁻³ |
| 269/11 | 24.4545454545 | 5.1 × 10⁻⁴ |
| 4,255/174 | 24.4540229885 | 1.6 × 10⁻⁵ |
| 8,779/359 | 24.4540389972 | 4.8 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 598y² = 1. Its smallest solution in positive whole numbers is x = 1,574,351, y = 64,380.
√598 in geometry and everyday measurements
- A square garage floor of 598 square feet measures about 24.45 ft (24 ft 5 in) per side, and its corner-to-corner diagonal is √1196 ≈ 34.6 ft.
- 598 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √598 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 6 × 11 × 21 box, because 6² + 11² + 21² = 598.
Square roots near √598 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √595 | √595 | 24.3926 | No |
| √596 | 2√149 | 24.4131 | No |
| √597 | √597 | 24.4336 | No |
| √598 | √598 | 24.4540 | No |
| √599 | √599 | 24.4745 | No |
| √600 | 10√6 | 24.4949 | No |
| √601 | √601 | 24.5153 | No |
- The cube root of 598 is about 8.424945.
- Squaring undoes the root: (√598)² = 598, while 598² = 357,604 — the number whose square root is 598.
Frequently asked questions
What is the square root of 598?
The square root of 598 is √598, about 24.4540385213. The negative root, −24.454039, also squares to 598.
Is the square root of 598 rational or irrational?
Irrational. 598 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 √598 be simplified?
No. 598 = 2 × 13 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √598 rounded to two decimal places?
√598 ≈ 24.45 to two decimal places (24.5 to one, 24.454 to three). Check: 24.45² = 597.8025, close to 598.