√497 at a glance
- Exact value
- √497
- Decimal (10 places)
- 22.2934968096
- Rounded
- 22.3 · 22.29 · 22.293
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.293497
- Prime factorization
- 7 × 71
- Cube root
- 7.921099
How to simplify √497
The prime factorization of 497 is 7 × 71. Every prime appears only once, so there is no pair to bring outside the radical — √497 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 497, 7 and 71 appear an odd number of times, so √497 is irrational and 22.2934968096 is a rounded value.
Where √497 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √497 lies between 22 and 23. 497 is 13 above 484 and 32 below 529, so the root is closer to 22.
- Straight line between 484 and 529: 22.2889 (0.02% low)
- Tangent from 22, i.e. 22 + 13 ÷ 44: 22.2955 (0.01% high)
- Tangent from 23, i.e. 23 − 32 ÷ 46: 22.3043 (0.05% high)
For √497 the tangent at 22 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 497 is just 13 above 484.
Finding √497 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 497: following the tangent line down to zero simplifies to averaging x with 497 ÷ x.
Start from the nearest whole number, 22 (22² = 484):
| Step | Guess x | 497 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 22.5909090909 | 22.2954545455 | 2 |
| 2 | 22.2954545455 | 22.2915392457 | 22.2934968956 | 7 |
| 3 | 22.2934968956 | 22.2934967237 | 22.2934968096 | 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 √497 = 22.2934968096 to every decimal shown.
√497 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √497 the pattern is [22; 3, 2, 2, 5, 6, 5, 2, 2, 3, 44] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √497 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 | 2.9 × 10⁻¹ |
| 67/3 | 22.3333333333 | 4.0 × 10⁻² |
| 156/7 | 22.2857142857 | 7.8 × 10⁻³ |
| 379/17 | 22.2941176471 | 6.2 × 10⁻⁴ |
| 2,051/92 | 22.2934782609 | 1.9 × 10⁻⁵ |
| 12,685/569 | 22.2934973638 | 5.5 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 497y² = 1. Its smallest solution in positive whole numbers is x = 1,201,887, y = 53,912.
√497 in geometry and everyday measurements
- A square garage floor of 497 square feet measures about 22.29 ft (22 ft 4 in) per side, and its corner-to-corner diagonal is √994 ≈ 31.5 ft.
- 497 is not a sum of two whole-number squares — the prime factor 7 and 71 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √497 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 3 × 22 box, because 2² + 3² + 22² = 497.
Square roots near √497 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √494 | √494 | 22.2261 | No |
| √495 | 3√55 | 22.2486 | No |
| √496 | 4√31 | 22.2711 | No |
| √497 | √497 | 22.2935 | No |
| √498 | √498 | 22.3159 | No |
| √499 | √499 | 22.3383 | No |
| √500 | 10√5 | 22.3607 | No |
- The cube root of 497 is about 7.921099.
- Squaring undoes the root: (√497)² = 497, while 497² = 247,009 — the number whose square root is 497.
Frequently asked questions
What is the square root of 497?
The square root of 497 is √497, about 22.2934968096. The negative root, −22.293497, also squares to 497.
Is the square root of 497 rational or irrational?
Irrational. 497 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 √497 be simplified?
No. 497 = 7 × 71 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √497 rounded to two decimal places?
√497 ≈ 22.29 to two decimal places (22.3 to one, 22.293 to three). Check: 22.29² = 496.8441, close to 497.