√486 at a glance
- Exact value
- 9√6
- Decimal (10 places)
- 22.0454076850
- Rounded
- 22.0 · 22.05 · 22.045
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.045408
- Prime factorization
- 2 × 3⁵
- Cube root
- 7.862224
How to simplify √486
Look for the largest perfect square that divides 486. Here it is 81 (9²), because 486 = 81 × 6 and 6 has no square factor left:
The prime factorization tells the same story: 486 = 2 × 3⁵. Each pair of equal primes leaves the radical as one factor, so 3² comes out and 2 × 3 stays inside.
486 has 2 square factors (9 and 81). Starting with a smaller one still works but takes more rounds: √486 = 3√54, and √54 can be simplified again. Using 81 straight away finishes in one step.
Check: (9√6)² = 9² × 6 = 81 × 6 = 486. As a decimal, 9√6 = 9 × 2.4494897428 ≈ 22.0454076850.
Where √486 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √486 lies between 22 and 23. 486 is 2 above 484 and 43 below 529, so the root is closer to 22.
- Straight line between 484 and 529: 22.0444 (0% low)
- Tangent from 22, i.e. 22 + 2 ÷ 44: 22.0455 (0% high)
- Tangent from 23, i.e. 23 − 43 ÷ 46: 22.0652 (0.09% high)
For √486 the tangent at 22 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 486 is just 2 above 484.
Finding √486 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 | 486 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 22.0909090909 | 22.0454545455 | 4 |
| 2 | 22.0454545455 | 22.0453608247 | 22.0454076851 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √486 = 22.0454076850 to every decimal shown.
√486 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √486 the pattern is [22; 22, 44] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √486 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 | 4.5 × 10⁻² |
| 485/22 | 22.0454545455 | 4.7 × 10⁻⁵ |
| 21,362/969 | 22.0454076367 | 4.8 × 10⁻⁸ |
| 470,449/21,340 | 22.0454076851 | < 10⁻¹⁰ |
| 20,721,118/939,929 | 22.0454076850 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 486y² = 1. Its smallest solution in positive whole numbers is x = 485, y = 22.
√486 in geometry and everyday measurements
- A square garage floor of 486 square feet measures about 22.05 ft (22 ft 1 in) per side, and its corner-to-corner diagonal is √972 ≈ 31.2 ft.
- 486 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √486 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 22 box, because 1² + 1² + 22² = 486.
- Since √486 = 9√6, a length of √486 is exactly 9 copies of the length √6 laid end to end.
Square roots near √486 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √483 | √483 | 21.9773 | No |
| √484 | 22 | 22.0000 | Yes |
| √485 | √485 | 22.0227 | No |
| √486 | 9√6 | 22.0454 | No |
| √487 | √487 | 22.0681 | No |
| √488 | 2√122 | 22.0907 | No |
| √489 | √489 | 22.1133 | No |
- The cube root of 486 is about 7.862224.
- Squaring undoes the root: (√486)² = 486, while 486² = 236,196 — the number whose square root is 486.
Frequently asked questions
What is the square root of 486?
The square root of 486 is 9√6 in simplest radical form, which is about 22.0454076850. The negative root, −22.045408, also squares to 486.
Is the square root of 486 rational or irrational?
Irrational. 486 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 √486 be simplified?
Yes. The largest perfect square dividing 486 is 81, so √486 = √81 × √6 = 9√6.
What is √486 rounded to two decimal places?
√486 ≈ 22.05 to two decimal places (22.0 to one, 22.045 to three). Check: 22.05² = 486.2025, close to 486.