√821 at a glance
- Exact value
- √821
- Decimal (10 places)
- 28.6530975638
- Rounded
- 28.7 · 28.65 · 28.653
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.653098
- Prime factorization
- 821
- Cube root
- 9.363705
How to simplify √821
821 is a prime number, so its only factors are 1 and 821. There is no perfect-square factor to pull out, which means √821 is already in its simplest radical form.
The square root of any prime is irrational. If √821 were a fraction a/b in lowest terms, then a² = 821b², so 821 would divide a — and then 821 would divide b too, contradicting “lowest terms.” That is why the decimal 28.6530975638 is only a rounded value.
Where √821 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √821 lies between 28 and 29. 821 is 37 above 784 and 20 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.6491 (0.01% low)
- Tangent from 28, i.e. 28 + 37 ÷ 56: 28.6607 (0.03% high)
- Tangent from 29, i.e. 29 − 20 ÷ 58: 28.6552 (0.01% high)
For √821 the tangent at 29 wins, missing by only 0.0021. Tangent estimates shine when the number sits close to a perfect square — here 821 is just 20 below 841.
Finding √821 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 821: following the tangent line down to zero simplifies to averaging x with 821 ÷ x.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 821 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.3103448276 | 28.6551724138 | 2 |
| 2 | 28.6551724138 | 28.6510228640 | 28.6530976389 | 7 |
| 3 | 28.6530976389 | 28.6530974887 | 28.6530975638 | 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 √821 = 28.6530975638 to every decimal shown.
√821 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √821 the pattern is [28; 1, 1, 1, 7, 1, 1, 10, 1, 13, 2, 2, 2, …] with the block of 29 terms after the semicolon repeating forever (only the first 12 of the 29 are shown). A pattern that never ends is one more proof that √821 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 | 6.5 × 10⁻¹ |
| 29/1 | 29.0000000000 | 3.5 × 10⁻¹ |
| 57/2 | 28.5000000000 | 1.5 × 10⁻¹ |
| 86/3 | 28.6666666667 | 1.4 × 10⁻² |
| 659/23 | 28.6521739130 | 9.2 × 10⁻⁴ |
| 745/26 | 28.6538461538 | 7.5 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 821y² = 1. Its smallest solution in positive whole numbers is x = 9,000,987,377,460,935,993,101,449, y = 314,136,625,452,886,403,879,740 — 25 digits for x, even though 821 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 2,121,436,703,918² − 821 × 74,038,651,465² = −1.
√821 in geometry and everyday measurements
- 821 square feet is 76.3 m². Laid out as a square — a small house footprint or a lot — it is about 28.65 ft (28 ft 8 in) on a side.
- 821 = 14² + 25², so by the Pythagorean theorem √821 is the diagonal of a 14 × 25 rectangle — and the distance between the points (0, 0) and (14, 25) on a grid.
Square roots near √821 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √818 | √818 | 28.6007 | No |
| √819 | 3√91 | 28.6182 | No |
| √820 | 2√205 | 28.6356 | No |
| √821 | √821 | 28.6531 | No |
| √822 | √822 | 28.6705 | No |
| √823 | √823 | 28.6880 | No |
| √824 | 2√206 | 28.7054 | No |
- The cube root of 821 is about 9.363705.
- Squaring undoes the root: (√821)² = 821, while 821² = 674,041 — the number whose square root is 821.
Frequently asked questions
What is the square root of 821?
The square root of 821 is √821, about 28.6530975638. The negative root, −28.653098, also squares to 821.
Is the square root of 821 rational or irrational?
Irrational. 821 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 √821 be simplified?
No. 821 is prime, so there is no perfect square to take out of the radical.
What is √821 rounded to two decimal places?
√821 ≈ 28.65 to two decimal places (28.7 to one, 28.653 to three). Check: 28.65² = 820.8225, close to 821.