√514 at a glance
- Exact value
- √514
- Decimal (10 places)
- 22.6715680975
- Rounded
- 22.7 · 22.67 · 22.672
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.671568
- Prime factorization
- 2 × 257
- Cube root
- 8.010403
How to simplify √514
The prime factorization of 514 is 2 × 257. Every prime appears only once, so there is no pair to bring outside the radical — √514 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 514, 2 and 257 appear an odd number of times, so √514 is irrational and 22.6715680975 is a rounded value.
Where √514 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √514 lies between 22 and 23. 514 is 30 above 484 and 15 below 529, so the root is closer to 23.
- Straight line between 484 and 529: 22.6667 (0.02% low)
- Tangent from 22, i.e. 22 + 30 ÷ 44: 22.6818 (0.05% high)
- Tangent from 23, i.e. 23 − 15 ÷ 46: 22.6739 (0.01% high)
For √514 the tangent at 23 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 514 is just 15 below 529.
Finding √514 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, 23 (23² = 529):
| Step | Guess x | 514 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 23.0000000000 | 22.3478260870 | 22.6739130435 | 2 |
| 2 | 22.6739130435 | 22.6692233941 | 22.6715682188 | 6 |
| 3 | 22.6715682188 | 22.6715679763 | 22.6715680975 | 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 √514 = 22.6715680975 to every decimal shown.
√514 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √514 the pattern is [22; 1, 2, 22, 2, 1, 44] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √514 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 | 6.7 × 10⁻¹ |
| 23/1 | 23.0000000000 | 3.3 × 10⁻¹ |
| 68/3 | 22.6666666667 | 4.9 × 10⁻³ |
| 1,519/67 | 22.6716417910 | 7.4 × 10⁻⁵ |
| 3,106/137 | 22.6715328467 | 3.5 × 10⁻⁵ |
| 4,625/204 | 22.6715686275 | 5.3 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 514y² = 1. Its smallest solution in positive whole numbers is x = 4,625, y = 204.
√514 in geometry and everyday measurements
- A square garage floor of 514 square feet measures about 22.67 ft (22 ft 8 in) per side, and its corner-to-corner diagonal is √1028 ≈ 32.1 ft.
- 514 = 15² + 17², so by the Pythagorean theorem √514 is the diagonal of a 15 × 17 rectangle — and the distance between the points (0, 0) and (15, 17) on a grid.
Square roots near √514 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √511 | √511 | 22.6053 | No |
| √512 | 16√2 | 22.6274 | No |
| √513 | 3√57 | 22.6495 | No |
| √514 | √514 | 22.6716 | No |
| √515 | √515 | 22.6936 | No |
| √516 | 2√129 | 22.7156 | No |
| √517 | √517 | 22.7376 | No |
- The cube root of 514 is about 8.010403.
- Squaring undoes the root: (√514)² = 514, while 514² = 264,196 — the number whose square root is 514.
Frequently asked questions
What is the square root of 514?
The square root of 514 is √514, about 22.6715680975. The negative root, −22.671568, also squares to 514.
Is the square root of 514 rational or irrational?
Irrational. 514 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 √514 be simplified?
No. 514 = 2 × 257 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √514 rounded to two decimal places?
√514 ≈ 22.67 to two decimal places (22.7 to one, 22.672 to three). Check: 22.67² = 513.9289, close to 514.