√801 at a glance
- Exact value
- 3√89
- Decimal (10 places)
- 28.3019433962
- Rounded
- 28.3 · 28.30 · 28.302
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.301943
- Prime factorization
- 3² × 89
- Cube root
- 9.287044
How to simplify √801
Look for the largest perfect square that divides 801. Here it is 9 (3²), because 801 = 9 × 89 and 89 has no square factor left:
The prime factorization tells the same story: 801 = 3² × 89. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 89 stays inside.
Check: (3√89)² = 3² × 89 = 9 × 89 = 801. As a decimal, 3√89 = 3 × 9.4339811321 ≈ 28.3019433962.
Where √801 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √801 lies between 28 and 29. 801 is 17 above 784 and 40 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.2982 (0.01% low)
- Tangent from 28, i.e. 28 + 17 ÷ 56: 28.3036 (0.01% high)
- Tangent from 29, i.e. 29 − 40 ÷ 58: 28.3103 (0.03% high)
For √801 the tangent at 28 wins, missing by only 0.0016. Tangent estimates shine when the number sits close to a perfect square — here 801 is just 17 above 784.
Finding √801 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 801: following the tangent line down to zero simplifies to averaging x with 801 ÷ x.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 801 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.6071428571 | 28.3035714286 | 2 |
| 2 | 28.3035714286 | 28.3003154574 | 28.3019434430 | 7 |
| 3 | 28.3019434430 | 28.3019433493 | 28.3019433962 | 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 √801 = 28.3019433962 to every decimal shown.
√801 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √801 the pattern is [28; 3, 3, 4, 1, 5, 2, 10, 1, 6, 6, 6, 1, …] with the block of 20 terms after the semicolon repeating forever (only the first 12 of the 20 are shown). A pattern that never ends is one more proof that √801 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 | 3.0 × 10⁻¹ |
| 85/3 | 28.3333333333 | 3.1 × 10⁻² |
| 283/10 | 28.3000000000 | 1.9 × 10⁻³ |
| 1,217/43 | 28.3023255814 | 3.8 × 10⁻⁴ |
| 1,500/53 | 28.3018867925 | 5.7 × 10⁻⁵ |
| 8,717/308 | 28.3019480519 | 4.7 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 801y² = 1. Its smallest solution in positive whole numbers is x = 500,002,000,001, y = 17,666,702,000.
√801 in geometry and everyday measurements
- 801 square feet is 74.4 m². Laid out as a square — a small house footprint or a lot — it is about 28.3 ft (28 ft 4 in) on a side.
- 801 = 15² + 24², so by the Pythagorean theorem √801 is the diagonal of a 15 × 24 rectangle — and the distance between the points (0, 0) and (15, 24) on a grid.
- Since √801 = 3√89, a length of √801 is exactly 3 copies of the length √89 laid end to end.
Square roots near √801 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √804 | 2√201 | 28.3549 | No |
- The cube root of 801 is about 9.287044.
- Squaring undoes the root: (√801)² = 801, while 801² = 641,601 — the number whose square root is 801.
Frequently asked questions
What is the square root of 801?
The square root of 801 is 3√89 in simplest radical form, which is about 28.3019433962. The negative root, −28.301943, also squares to 801.
Is the square root of 801 rational or irrational?
Irrational. 801 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 √801 be simplified?
Yes. The largest perfect square dividing 801 is 9, so √801 = √9 × √89 = 3√89.
What is √801 rounded to two decimal places?
√801 ≈ 28.30 to two decimal places (28.3 to one, 28.302 to three). Check: 28.30² = 800.89, close to 801.