√498 at a glance
- Exact value
- √498
- Decimal (10 places)
- 22.3159136044
- Rounded
- 22.3 · 22.32 · 22.316
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.315914
- Prime factorization
- 2 × 3 × 83
- Cube root
- 7.926408
How to simplify √498
The prime factorization of 498 is 2 × 3 × 83. Every prime appears only once, so there is no pair to bring outside the radical — √498 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 498, 2, 3 and 83 appear an odd number of times, so √498 is irrational and 22.3159136044 is a rounded value.
Where √498 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √498 lies between 22 and 23. 498 is 14 above 484 and 31 below 529, so the root is closer to 22.
- Straight line between 484 and 529: 22.3111 (0.02% low)
- Tangent from 22, i.e. 22 + 14 ÷ 44: 22.3182 (0.01% high)
- Tangent from 23, i.e. 23 − 31 ÷ 46: 22.3261 (0.05% high)
For √498 the tangent at 22 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 498 is just 14 above 484.
Finding √498 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, 22 (22² = 484):
| Step | Guess x | 498 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 22.6363636364 | 22.3181818182 | 2 |
| 2 | 22.3181818182 | 22.3136456212 | 22.3159137197 | 6 |
| 3 | 22.3159137197 | 22.3159134892 | 22.3159136044 | 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 √498 = 22.3159136044 to every decimal shown.
√498 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √498 the pattern is [22; 3, 6, 22, 6, 3, 44] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √498 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 |
|---|---|---|
| 22/1 | 22.0000000000 | 3.2 × 10⁻¹ |
| 67/3 | 22.3333333333 | 1.7 × 10⁻² |
| 424/19 | 22.3157894737 | 1.2 × 10⁻⁴ |
| 9,395/421 | 22.3159144893 | 8.8 × 10⁻⁷ |
| 56,794/2,545 | 22.3159135560 | 4.8 × 10⁻⁸ |
| 179,777/8,056 | 22.3159136048 | 3.5 × 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 498y² = 1. Its smallest solution in positive whole numbers is x = 179,777, y = 8,056.
√498 in geometry and everyday measurements
- A square garage floor of 498 square feet measures about 22.32 ft (22 ft 4 in) per side, and its corner-to-corner diagonal is √996 ≈ 31.6 ft.
- 498 is not a sum of two whole-number squares — the prime factor 3 and 83 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √498 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 11 × 19 box, because 4² + 11² + 19² = 498.
Square roots near √498 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √495 | 3√55 | 22.2486 | No |
| √496 | 4√31 | 22.2711 | No |
| √497 | √497 | 22.2935 | No |
| √498 | √498 | 22.3159 | No |
| √499 | √499 | 22.3383 | No |
| √500 | 10√5 | 22.3607 | No |
| √501 | √501 | 22.3830 | No |
- The cube root of 498 is about 7.926408.
- Squaring undoes the root: (√498)² = 498, while 498² = 248,004 — the number whose square root is 498.
Frequently asked questions
What is the square root of 498?
The square root of 498 is √498, about 22.3159136044. The negative root, −22.315914, also squares to 498.
Is the square root of 498 rational or irrational?
Irrational. 498 is not a perfect square — it falls between 484 and 529 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √498 be simplified?
No. 498 = 2 × 3 × 83 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √498 rounded to two decimal places?
√498 ≈ 22.32 to two decimal places (22.3 to one, 22.316 to three). Check: 22.32² = 498.1824, close to 498.