√800 at a glance
- Exact value
- 20√2
- Decimal (10 places)
- 28.2842712475
- Rounded
- 28.3 · 28.28 · 28.284
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.284271
- Prime factorization
- 2⁵ × 5²
- Cube root
- 9.283178
How to simplify √800
Look for the largest perfect square that divides 800. Here it is 400 (20²), because 800 = 400 × 2 and 2 has no square factor left:
The prime factorization tells the same story: 800 = 2⁵ × 5². Each pair of equal primes leaves the radical as one factor, so 2² × 5 comes out and 2 stays inside.
800 has 5 square factors (4, 16, 25, 100 and 400). Starting with a smaller one still works but takes more rounds: √800 = 2√200, and √200 can be simplified again. Using 400 straight away finishes in one step.
Check: (20√2)² = 20² × 2 = 400 × 2 = 800. As a decimal, 20√2 = 20 × 1.4142135624 ≈ 28.2842712475.
Where √800 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √800 lies between 28 and 29. 800 is 16 above 784 and 41 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.2807 (0.01% low)
- Tangent from 28, i.e. 28 + 16 ÷ 56: 28.2857 (0.01% high)
- Tangent from 29, i.e. 29 − 41 ÷ 58: 28.2931 (0.03% high)
For √800 the tangent at 28 wins, missing by only 0.0014. Tangent estimates shine when the number sits close to a perfect square — here 800 is just 16 above 784.
Finding √800 with the Babylonian method
Picture a rectangle with an area of 800 and one side x; the other side must be 800 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √800.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 800 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.5714285714 | 28.2857142857 | 2 |
| 2 | 28.2857142857 | 28.2828282828 | 28.2842712843 | 7 |
| 3 | 28.2842712843 | 28.2842712107 | 28.2842712475 | 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 √800 = 28.2842712475 to every decimal shown.
√800 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √800 the pattern is [28; 3, 1, 1, 13, 1, 1, 3, 56] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √800 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 | 2.8 × 10⁻¹ |
| 85/3 | 28.3333333333 | 4.9 × 10⁻² |
| 113/4 | 28.2500000000 | 3.4 × 10⁻² |
| 198/7 | 28.2857142857 | 1.4 × 10⁻³ |
| 2,687/95 | 28.2842105263 | 6.1 × 10⁻⁵ |
| 2,885/102 | 28.2843137255 | 4.2 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 800y² = 1. Its smallest solution in positive whole numbers is x = 19,601, y = 693.
√800 in geometry and everyday measurements
- 800 square feet is 74.3 m². Laid out as a square — a small house footprint or a lot — it is about 28.28 ft (28 ft 3 in) on a side.
- 800 = 4² + 28² = 20² + 20², so by the Pythagorean theorem √800 is the diagonal of rectangles measuring 4 × 28 and 20 × 20 — and the distance between the points (0, 0) and (4, 28) on a grid.
- Since √800 = 20√2, a length of √800 is exactly 20 copies of the length √2 laid end to end.
Square roots near √800 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √797 | √797 | 28.2312 | No |
| √798 | √798 | 28.2489 | No |
| √799 | √799 | 28.2666 | No |
| √800 | 20√2 | 28.2843 | No |
| √801 | 3√89 | 28.3019 | No |
| √802 | √802 | 28.3196 | No |
| √803 | √803 | 28.3373 | No |
- The cube root of 800 is about 9.283178.
- Dividing by 100 divides the root by 10: √8 = √800 ÷ 10 ≈ 2.82842712.
Frequently asked questions
What is the square root of 800?
The square root of 800 is 20√2 in simplest radical form, which is about 28.2842712475. The negative root, −28.284271, also squares to 800.
Is the square root of 800 rational or irrational?
Irrational. 800 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 √800 be simplified?
Yes. The largest perfect square dividing 800 is 400, so √800 = √400 × √2 = 20√2.
What is √800 rounded to two decimal places?
√800 ≈ 28.28 to two decimal places (28.3 to one, 28.284 to three). Check: 28.28² = 799.7584, close to 800.