√501 at a glance
- Exact value
- √501
- Decimal (10 places)
- 22.3830292856
- Rounded
- 22.4 · 22.38 · 22.383
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.383029
- Prime factorization
- 3 × 167
- Cube root
- 7.942293
How to simplify √501
The prime factorization of 501 is 3 × 167. Every prime appears only once, so there is no pair to bring outside the radical — √501 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 501, 3 and 167 appear an odd number of times, so √501 is irrational and 22.3830292856 is a rounded value.
Where √501 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √501 lies between 22 and 23. 501 is 17 above 484 and 28 below 529, so the root is closer to 22.
- Straight line between 484 and 529: 22.3778 (0.02% low)
- Tangent from 22, i.e. 22 + 17 ÷ 44: 22.3864 (0.01% high)
- Tangent from 23, i.e. 23 − 28 ÷ 46: 22.3913 (0.04% high)
For √501 the tangent at 22 wins, missing by only 0.0033. Tangent estimates shine when the number sits close to a perfect square — here 501 is just 17 above 484.
Finding √501 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 501: following the tangent line down to zero simplifies to averaging x with 501 ÷ x.
Start from the nearest whole number, 22 (22² = 484):
| Step | Guess x | 501 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 22.7727272727 | 22.3863636364 | 2 |
| 2 | 22.3863636364 | 22.3796954315 | 22.3830295339 | 6 |
| 3 | 22.3830295339 | 22.3830290373 | 22.3830292856 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √501 = 22.3830292856 to every decimal shown.
√501 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √501 the pattern is [22; 2, 1, 1, 1, 1, 3, 8, 1, 2, 10, 1, 5, …] with the block of 28 terms after the semicolon repeating forever (only the first 12 of the 28 are shown). A pattern that never ends is one more proof that √501 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 | 3.8 × 10⁻¹ |
| 45/2 | 22.5000000000 | 1.2 × 10⁻¹ |
| 67/3 | 22.3333333333 | 5.0 × 10⁻² |
| 112/5 | 22.4000000000 | 1.7 × 10⁻² |
| 179/8 | 22.3750000000 | 8.0 × 10⁻³ |
| 291/13 | 22.3846153846 | 1.6 × 10⁻³ |
The same fractions solve Pell’s equation, x² − 501y² = 1. Its smallest solution in positive whole numbers is x = 11,242,731,902,975, y = 502,288,218,432 — 14 digits for x, even though 501 is small, which is what makes Pell’s equation famous.
√501 in geometry and everyday measurements
- A square garage floor of 501 square feet measures about 22.38 ft (22 ft 5 in) per side, and its corner-to-corner diagonal is √1002 ≈ 31.7 ft.
- 501 is not a sum of two whole-number squares — the prime factor 3 and 167 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √501 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 22 box, because 1² + 4² + 22² = 501.
Square roots near √501 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √498 | √498 | 22.3159 | No |
| √499 | √499 | 22.3383 | No |
| √500 | 10√5 | 22.3607 | No |
| √501 | √501 | 22.3830 | No |
| √502 | √502 | 22.4054 | No |
| √503 | √503 | 22.4277 | No |
| √504 | 6√14 | 22.4499 | No |
- The cube root of 501 is about 7.942293.
- Squaring undoes the root: (√501)² = 501, while 501² = 251,001 — the number whose square root is 501.
Frequently asked questions
What is the square root of 501?
The square root of 501 is √501, about 22.3830292856. The negative root, −22.383029, also squares to 501.
Is the square root of 501 rational or irrational?
Irrational. 501 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 √501 be simplified?
No. 501 = 3 × 167 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √501 rounded to two decimal places?
√501 ≈ 22.38 to two decimal places (22.4 to one, 22.383 to three). Check: 22.38² = 500.8644, close to 501.