√490 at a glance
- Exact value
- 7√10
- Decimal (10 places)
- 22.1359436212
- Rounded
- 22.1 · 22.14 · 22.136
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.135944
- Prime factorization
- 2 × 5 × 7²
- Cube root
- 7.883735
How to simplify √490
Look for the largest perfect square that divides 490. Here it is 49 (7²), because 490 = 49 × 10 and 10 has no square factor left:
The prime factorization tells the same story: 490 = 2 × 5 × 7². Each pair of equal primes leaves the radical as one factor, so 7 comes out and 2 × 5 stays inside.
Check: (7√10)² = 7² × 10 = 49 × 10 = 490. As a decimal, 7√10 = 7 × 3.1622776602 ≈ 22.1359436212.
Where √490 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √490 lies between 22 and 23. 490 is 6 above 484 and 39 below 529, so the root is closer to 22.
- Straight line between 484 and 529: 22.1333 (0.01% low)
- Tangent from 22, i.e. 22 + 6 ÷ 44: 22.1364 (0% high)
- Tangent from 23, i.e. 23 − 39 ÷ 46: 22.1522 (0.07% high)
For √490 the tangent at 22 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 490 is just 6 above 484.
Finding √490 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 | 490 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 22.2727272727 | 22.1363636364 | 3 |
| 2 | 22.1363636364 | 22.1355236140 | 22.1359436252 | 8 |
| 3 | 22.1359436252 | 22.1359436172 | 22.1359436212 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √490 = 22.1359436212 to every decimal shown.
√490 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √490 the pattern is [22; 7, 2, 1, 4, 4, 4, 1, 2, 7, 44] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √490 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 | 1.4 × 10⁻¹ |
| 155/7 | 22.1428571429 | 6.9 × 10⁻³ |
| 332/15 | 22.1333333333 | 2.6 × 10⁻³ |
| 487/22 | 22.1363636364 | 4.2 × 10⁻⁴ |
| 2,280/103 | 22.1359223301 | 2.1 × 10⁻⁵ |
| 9,607/434 | 22.1359447005 | 1.1 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 490y² = 1. Its smallest solution in positive whole numbers is x = 1,039,681, y = 46,968.
√490 in geometry and everyday measurements
- A square garage floor of 490 square feet measures about 22.14 ft (22 ft 2 in) per side, and its corner-to-corner diagonal is √980 ≈ 31.3 ft.
- 490 = 7² + 21², so by the Pythagorean theorem √490 is the diagonal of a 7 × 21 rectangle — and the distance between the points (0, 0) and (7, 21) on a grid.
- Since √490 = 7√10, a length of √490 is exactly 7 copies of the length √10 laid end to end.
Square roots near √490 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √487 | √487 | 22.0681 | No |
| √488 | 2√122 | 22.0907 | No |
| √489 | √489 | 22.1133 | No |
| √490 | 7√10 | 22.1359 | No |
| √491 | √491 | 22.1585 | No |
| √492 | 2√123 | 22.1811 | No |
| √493 | √493 | 22.2036 | No |
- The cube root of 490 is about 7.883735.
- Squaring undoes the root: (√490)² = 490, while 490² = 240,100 — the number whose square root is 490.
Frequently asked questions
What is the square root of 490?
The square root of 490 is 7√10 in simplest radical form, which is about 22.1359436212. The negative root, −22.135944, also squares to 490.
Is the square root of 490 rational or irrational?
Irrational. 490 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 √490 be simplified?
Yes. The largest perfect square dividing 490 is 49, so √490 = √49 × √10 = 7√10.
What is √490 rounded to two decimal places?
√490 ≈ 22.14 to two decimal places (22.1 to one, 22.136 to three). Check: 22.14² = 490.1796, close to 490.