√820 at a glance
- Exact value
- 2√205
- Decimal (10 places)
- 28.6356421266
- Rounded
- 28.6 · 28.64 · 28.636
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.635642
- Prime factorization
- 2² × 5 × 41
- Cube root
- 9.359902
How to simplify √820
Look for the largest perfect square that divides 820. Here it is 4 (2²), because 820 = 4 × 205 and 205 has no square factor left:
The prime factorization tells the same story: 820 = 2² × 5 × 41. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 5 × 41 stays inside.
Check: (2√205)² = 2² × 205 = 4 × 205 = 820. As a decimal, 2√205 = 2 × 14.3178210633 ≈ 28.6356421266.
Where √820 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √820 lies between 28 and 29. 820 is 36 above 784 and 21 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.6316 (0.01% low)
- Tangent from 28, i.e. 28 + 36 ÷ 56: 28.6429 (0.03% high)
- Tangent from 29, i.e. 29 − 21 ÷ 58: 28.6379 (0.01% high)
For √820 the tangent at 29 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 820 is just 21 below 841.
Finding √820 with the Babylonian method
Picture a rectangle with an area of 820 and one side x; the other side must be 820 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √820.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 820 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.2758620690 | 28.6379310345 | 2 |
| 2 | 28.6379310345 | 28.6333534016 | 28.6356422180 | 7 |
| 3 | 28.6356422180 | 28.6356420351 | 28.6356421266 | 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 √820 = 28.6356421266 to every decimal shown.
√820 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √820 the pattern is [28; 1, 1, 1, 2, 1, 10, 1, 2, 1, 1, 1, 56] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √820 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 | 6.4 × 10⁻¹ |
| 29/1 | 29.0000000000 | 3.6 × 10⁻¹ |
| 57/2 | 28.5000000000 | 1.4 × 10⁻¹ |
| 86/3 | 28.6666666667 | 3.1 × 10⁻² |
| 229/8 | 28.6250000000 | 1.1 × 10⁻² |
| 315/11 | 28.6363636364 | 7.2 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 820y² = 1. Its smallest solution in positive whole numbers is x = 39,689, y = 1,386.
√820 in geometry and everyday measurements
- 820 square feet is 76.2 m². Laid out as a square — a small house footprint or a lot — it is about 28.64 ft (28 ft 8 in) on a side.
- 820 = 6² + 28² = 12² + 26², so by the Pythagorean theorem √820 is the diagonal of rectangles measuring 6 × 28 and 12 × 26 — and the distance between the points (0, 0) and (6, 28) on a grid.
- Since √820 = 2√205, a length of √820 is exactly 2 copies of the length √205 laid end to end.
Square roots near √820 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √817 | √817 | 28.5832 | No |
| √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 |
- The cube root of 820 is about 9.359902.
- Because 820 = 4 × 205, the root is twice √205: 2 × 14.317821 ≈ 28.635642.
Frequently asked questions
What is the square root of 820?
The square root of 820 is 2√205 in simplest radical form, which is about 28.6356421266. The negative root, −28.635642, also squares to 820.
Is the square root of 820 rational or irrational?
Irrational. 820 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 √820 be simplified?
Yes. The largest perfect square dividing 820 is 4, so √820 = √4 × √205 = 2√205.
What is √820 rounded to two decimal places?
√820 ≈ 28.64 to two decimal places (28.6 to one, 28.636 to three). Check: 28.64² = 820.2496, close to 820.