√826 at a glance
- Exact value
- √826
- Decimal (10 places)
- 28.7402157264
- Rounded
- 28.7 · 28.74 · 28.740
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.740216
- Prime factorization
- 2 × 7 × 59
- Cube root
- 9.382675
How to simplify √826
The prime factorization of 826 is 2 × 7 × 59. Every prime appears only once, so there is no pair to bring outside the radical — √826 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 826, 2, 7 and 59 appear an odd number of times, so √826 is irrational and 28.7402157264 is a rounded value.
Where √826 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √826 lies between 28 and 29. 826 is 42 above 784 and 15 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.7368 (0.01% low)
- Tangent from 28, i.e. 28 + 42 ÷ 56: 28.7500 (0.03% high)
- Tangent from 29, i.e. 29 − 15 ÷ 58: 28.7414 (0% high)
For √826 the tangent at 29 wins, missing by only 0.0012. Tangent estimates shine when the number sits close to a perfect square — here 826 is just 15 below 841.
Finding √826 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, 29 (29² = 841):
| Step | Guess x | 826 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.4827586207 | 28.7413793103 | 2 |
| 2 | 28.7413793103 | 28.7390521896 | 28.7402157500 | 7 |
| 3 | 28.7402157500 | 28.7402157028 | 28.7402157264 | 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 √826 = 28.7402157264 to every decimal shown.
√826 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √826 the pattern is [28; 1, 2, 1, 5, 1, 1, 1, 3, 5, 2, 9, 8, …] with the block of 24 terms after the semicolon repeating forever (only the first 12 of the 24 are shown). A pattern that never ends is one more proof that √826 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 |
|---|---|---|
| 28/1 | 28.0000000000 | 7.4 × 10⁻¹ |
| 29/1 | 29.0000000000 | 2.6 × 10⁻¹ |
| 86/3 | 28.6666666667 | 7.4 × 10⁻² |
| 115/4 | 28.7500000000 | 9.8 × 10⁻³ |
| 661/23 | 28.7391304348 | 1.1 × 10⁻³ |
| 776/27 | 28.7407407407 | 5.3 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 826y² = 1. Its smallest solution in positive whole numbers is x = 222,239,304,685, y = 7,732,694,382.
√826 in geometry and everyday measurements
- 826 square feet is 76.7 m². Laid out as a square — a small house footprint or a lot — it is about 28.74 ft (28 ft 9 in) on a side.
- 826 is not a sum of two whole-number squares — the prime factor 7 and 59 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √826 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 9 × 27 box, because 4² + 9² + 27² = 826.
Square roots near √826 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √823 | √823 | 28.6880 | No |
| √824 | 2√206 | 28.7054 | No |
| √825 | 5√33 | 28.7228 | No |
| √826 | √826 | 28.7402 | No |
| √827 | √827 | 28.7576 | No |
| √828 | 6√23 | 28.7750 | No |
| √829 | √829 | 28.7924 | No |
- The cube root of 826 is about 9.382675.
- Squaring undoes the root: (√826)² = 826, while 826² = 682,276 — the number whose square root is 826.
Frequently asked questions
What is the square root of 826?
The square root of 826 is √826, about 28.7402157264. The negative root, −28.740216, also squares to 826.
Is the square root of 826 rational or irrational?
Irrational. 826 is not a perfect square — it falls between 784 and 841 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √826 be simplified?
No. 826 = 2 × 7 × 59 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √826 rounded to two decimal places?
√826 ≈ 28.74 to two decimal places (28.7 to one, 28.740 to three). Check: 28.74² = 825.9876, close to 826.