√491 at a glance
- Exact value
- √491
- Decimal (10 places)
- 22.1585198062
- Rounded
- 22.2 · 22.16 · 22.159
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.158520
- Prime factorization
- 491
- Cube root
- 7.889095
How to simplify √491
491 is a prime number, so its only factors are 1 and 491. There is no perfect-square factor to pull out, which means √491 is already in its simplest radical form.
The square root of any prime is irrational. If √491 were a fraction a/b in lowest terms, then a² = 491b², so 491 would divide a — and then 491 would divide b too, contradicting “lowest terms.” That is why the decimal 22.1585198062 is only a rounded value.
Where √491 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √491 lies between 22 and 23. 491 is 7 above 484 and 38 below 529, so the root is closer to 22.
- Straight line between 484 and 529: 22.1556 (0.01% low)
- Tangent from 22, i.e. 22 + 7 ÷ 44: 22.1591 (0% high)
- Tangent from 23, i.e. 23 − 38 ÷ 46: 22.1739 (0.07% high)
For √491 the tangent at 22 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 491 is just 7 above 484.
Finding √491 with the Babylonian method
If a guess is too big, 491 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√491) in one step.
Start from the nearest whole number, 22 (22² = 484):
| Step | Guess x | 491 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 22.3181818182 | 22.1590909091 | 3 |
| 2 | 22.1590909091 | 22.1579487179 | 22.1585198135 | 8 |
| 3 | 22.1585198135 | 22.1585197988 | 22.1585198062 | 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 √491 = 22.1585198062 to every decimal shown.
√491 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √491 the pattern is [22; 6, 3, 4, 8, 1, 1, 1, 2, 1, 1, 21, 1, …] 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 √491 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 | 1.6 × 10⁻¹ |
| 133/6 | 22.1666666667 | 8.1 × 10⁻³ |
| 421/19 | 22.1578947368 | 6.3 × 10⁻⁴ |
| 1,817/82 | 22.1585365854 | 1.7 × 10⁻⁵ |
| 14,957/675 | 22.1585185185 | 1.3 × 10⁻⁶ |
| 16,774/757 | 22.1585204756 | 6.7 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 491y² = 1. Its smallest solution in positive whole numbers is x = 93,628,044,170, y = 4,225,374,483.
√491 in geometry and everyday measurements
- A square garage floor of 491 square feet measures about 22.16 ft (22 ft 2 in) per side, and its corner-to-corner diagonal is √982 ≈ 31.3 ft.
- 491 is not a sum of two whole-number squares — 491 is itself a prime that is one less than a multiple of 4, which rules that out — so √491 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 7 × 21 box, because 1² + 7² + 21² = 491.
Square roots near √491 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √494 | √494 | 22.2261 | No |
- The cube root of 491 is about 7.889095.
- Squaring undoes the root: (√491)² = 491, while 491² = 241,081 — the number whose square root is 491.
Frequently asked questions
What is the square root of 491?
The square root of 491 is √491, about 22.1585198062. The negative root, −22.158520, also squares to 491.
Is the square root of 491 rational or irrational?
Irrational. 491 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 √491 be simplified?
No. 491 is prime, so there is no perfect square to take out of the radical.
What is √491 rounded to two decimal places?
√491 ≈ 22.16 to two decimal places (22.2 to one, 22.159 to three). Check: 22.16² = 491.0656, close to 491.