√489 at a glance
- Exact value
- √489
- Decimal (10 places)
- 22.1133443875
- Rounded
- 22.1 · 22.11 · 22.113
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.113344
- Prime factorization
- 3 × 163
- Cube root
- 7.878368
How to simplify √489
The prime factorization of 489 is 3 × 163. Every prime appears only once, so there is no pair to bring outside the radical — √489 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 489, 3 and 163 appear an odd number of times, so √489 is irrational and 22.1133443875 is a rounded value.
Where √489 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √489 lies between 22 and 23. 489 is 5 above 484 and 40 below 529, so the root is closer to 22.
- Straight line between 484 and 529: 22.1111 (0.01% low)
- Tangent from 22, i.e. 22 + 5 ÷ 44: 22.1136 (0% high)
- Tangent from 23, i.e. 23 − 40 ÷ 46: 22.1304 (0.08% high)
For √489 the tangent at 22 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 489 is just 5 above 484.
Finding √489 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 489: following the tangent line down to zero simplifies to averaging x with 489 ÷ x.
Start from the nearest whole number, 22 (22² = 484):
| Step | Guess x | 489 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 22.2272727273 | 22.1136363636 | 3 |
| 2 | 22.1136363636 | 22.1130524152 | 22.1133443894 | 8 |
| 3 | 22.1133443894 | 22.1133443856 | 22.1133443875 | 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 √489 = 22.1133443875 to every decimal shown.
√489 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √489 the pattern is [22; 8, 1, 4, 1, 1, 1, 3, 2, 1, 2, 14, 2, …] with the block of 22 terms after the semicolon repeating forever (only the first 12 of the 22 are shown). A pattern that never ends is one more proof that √489 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 | 1.1 × 10⁻¹ |
| 177/8 | 22.1250000000 | 1.2 × 10⁻² |
| 199/9 | 22.1111111111 | 2.2 × 10⁻³ |
| 973/44 | 22.1136363636 | 2.9 × 10⁻⁴ |
| 1,172/53 | 22.1132075472 | 1.4 × 10⁻⁴ |
| 2,145/97 | 22.1134020619 | 5.8 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 489y² = 1. Its smallest solution in positive whole numbers is x = 7,592,629,975, y = 343,350,596.
√489 in geometry and everyday measurements
- A square garage floor of 489 square feet measures about 22.11 ft (22 ft 1 in) per side, and its corner-to-corner diagonal is √978 ≈ 31.3 ft.
- 489 is not a sum of two whole-number squares — the prime factor 3 and 163 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √489 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 22 box, because 1² + 2² + 22² = 489.
Square roots near √489 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √486 | 9√6 | 22.0454 | No |
| √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 |
- The cube root of 489 is about 7.878368.
- Squaring undoes the root: (√489)² = 489, while 489² = 239,121 — the number whose square root is 489.
Frequently asked questions
What is the square root of 489?
The square root of 489 is √489, about 22.1133443875. The negative root, −22.113344, also squares to 489.
Is the square root of 489 rational or irrational?
Irrational. 489 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 √489 be simplified?
No. 489 = 3 × 163 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √489 rounded to two decimal places?
√489 ≈ 22.11 to two decimal places (22.1 to one, 22.113 to three). Check: 22.11² = 488.8521, close to 489.