√526 at a glance
- Exact value
- √526
- Decimal (10 places)
- 22.9346898824
- Rounded
- 22.9 · 22.93 · 22.935
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.934690
- Prime factorization
- 2 × 263
- Cube root
- 8.072262
How to simplify √526
The prime factorization of 526 is 2 × 263. Every prime appears only once, so there is no pair to bring outside the radical — √526 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 526, 2 and 263 appear an odd number of times, so √526 is irrational and 22.9346898824 is a rounded value.
Where √526 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √526 lies between 22 and 23. 526 is 42 above 484 and 3 below 529, so the root is closer to 23.
- Straight line between 484 and 529: 22.9333 (0.01% low)
- Tangent from 22, i.e. 22 + 42 ÷ 44: 22.9545 (0.09% high)
- Tangent from 23, i.e. 23 − 3 ÷ 46: 22.9348 (0% high)
For √526 the tangent at 23 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 526 is just 3 below 529.
Finding √526 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 | 526 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 23.0000000000 | 22.8695652174 | 22.9347826087 | 4 |
| 2 | 22.9347826087 | 22.9345971564 | 22.9346898825 | 9 |
| 3 | 22.9346898825 | 22.9346898822 | 22.9346898824 | all 10 shown |
The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √526 = 22.9346898824 to every decimal shown.
√526 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √526 the pattern is [22; 1, 14, 3, 4, 1, 3, 2, 1, 3, 1, 8, 2, …] with the block of 40 terms after the semicolon repeating forever (only the first 12 of the 40 are shown). A pattern that never ends is one more proof that √526 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 | 9.3 × 10⁻¹ |
| 23/1 | 23.0000000000 | 6.5 × 10⁻² |
| 344/15 | 22.9333333333 | 1.4 × 10⁻³ |
| 1,055/46 | 22.9347826087 | 9.3 × 10⁻⁵ |
| 4,564/199 | 22.9346733668 | 1.7 × 10⁻⁵ |
| 5,619/245 | 22.9346938776 | 4.0 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 526y² = 1. Its smallest solution in positive whole numbers is x = 84,056,091,546,952,933,775, y = 3,665,019,757,324,295,532 — 20 digits for x, even though 526 is small, which is what makes Pell’s equation famous.
√526 in geometry and everyday measurements
- A square garage floor of 526 square feet measures about 22.93 ft (22 ft 11 in) per side, and its corner-to-corner diagonal is √1052 ≈ 32.4 ft.
- 526 is not a sum of two whole-number squares — the prime factor 263 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √526 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 9 × 21 box, because 2² + 9² + 21² = 526.
Square roots near √526 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √523 | √523 | 22.8692 | No |
| √524 | 2√131 | 22.8910 | No |
| √525 | 5√21 | 22.9129 | No |
| √526 | √526 | 22.9347 | No |
| √527 | √527 | 22.9565 | No |
| √528 | 4√33 | 22.9783 | No |
| √529 | 23 | 23.0000 | Yes |
- The cube root of 526 is about 8.072262.
- Squaring undoes the root: (√526)² = 526, while 526² = 276,676 — the number whose square root is 526.
Frequently asked questions
What is the square root of 526?
The square root of 526 is √526, about 22.9346898824. The negative root, −22.934690, also squares to 526.
Is the square root of 526 rational or irrational?
Irrational. 526 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 √526 be simplified?
No. 526 = 2 × 263 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √526 rounded to two decimal places?
√526 ≈ 22.93 to two decimal places (22.9 to one, 22.935 to three). Check: 22.93² = 525.7849, close to 526.