√494 at a glance
- Exact value
- √494
- Decimal (10 places)
- 22.2261107709
- Rounded
- 22.2 · 22.23 · 22.226
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.226111
- Prime factorization
- 2 × 13 × 19
- Cube root
- 7.905129
How to simplify √494
The prime factorization of 494 is 2 × 13 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √494 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 494, 2, 13 and 19 appear an odd number of times, so √494 is irrational and 22.2261107709 is a rounded value.
Where √494 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √494 lies between 22 and 23. 494 is 10 above 484 and 35 below 529, so the root is closer to 22.
- Straight line between 484 and 529: 22.2222 (0.02% low)
- Tangent from 22, i.e. 22 + 10 ÷ 44: 22.2273 (0.01% high)
- Tangent from 23, i.e. 23 − 35 ÷ 46: 22.2391 (0.06% high)
For √494 the tangent at 22 wins, missing by only 0.0012. Tangent estimates shine when the number sits close to a perfect square — here 494 is just 10 above 484.
Finding √494 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 | 494 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 22.4545454545 | 22.2272727273 | 2 |
| 2 | 22.2272727273 | 22.2249488753 | 22.2261108013 | 7 |
| 3 | 22.2261108013 | 22.2261107405 | 22.2261107709 | 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 √494 = 22.2261107709 to every decimal shown.
√494 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √494 the pattern is [22; 4, 2, 2, 1, 2, 1, 2, 2, 4, 44] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √494 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 | 2.3 × 10⁻¹ |
| 89/4 | 22.2500000000 | 2.4 × 10⁻² |
| 200/9 | 22.2222222222 | 3.9 × 10⁻³ |
| 489/22 | 22.2272727273 | 1.2 × 10⁻³ |
| 689/31 | 22.2258064516 | 3.0 × 10⁻⁴ |
| 1,867/84 | 22.2261904762 | 8.0 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 494y² = 1. Its smallest solution in positive whole numbers is x = 73,035, y = 3,286.
√494 in geometry and everyday measurements
- A square garage floor of 494 square feet measures about 22.23 ft (22 ft 3 in) per side, and its corner-to-corner diagonal is √988 ≈ 31.4 ft.
- 494 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √494 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 22 box, because 1² + 3² + 22² = 494.
Square roots near √494 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √491 | √491 | 22.1585 | No |
| √492 | 2√123 | 22.1811 | No |
| √493 | √493 | 22.2036 | No |
| √494 | √494 | 22.2261 | No |
| √495 | 3√55 | 22.2486 | No |
| √496 | 4√31 | 22.2711 | No |
| √497 | √497 | 22.2935 | No |
- The cube root of 494 is about 7.905129.
- Squaring undoes the root: (√494)² = 494, while 494² = 244,036 — the number whose square root is 494.
Frequently asked questions
What is the square root of 494?
The square root of 494 is √494, about 22.2261107709. The negative root, −22.226111, also squares to 494.
Is the square root of 494 rational or irrational?
Irrational. 494 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 √494 be simplified?
No. 494 = 2 × 13 × 19 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √494 rounded to two decimal places?
√494 ≈ 22.23 to two decimal places (22.2 to one, 22.226 to three). Check: 22.23² = 494.1729, close to 494.