√846 at a glance
- Exact value
- 3√94
- Decimal (10 places)
- 29.0860791445
- Rounded
- 29.1 · 29.09 · 29.086
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.086079
- Prime factorization
- 2 × 3² × 47
- Cube root
- 9.457800
How to simplify √846
Look for the largest perfect square that divides 846. Here it is 9 (3²), because 846 = 9 × 94 and 94 has no square factor left:
The prime factorization tells the same story: 846 = 2 × 3² × 47. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 47 stays inside.
Check: (3√94)² = 3² × 94 = 9 × 94 = 846. As a decimal, 3√94 = 3 × 9.6953597148 ≈ 29.0860791445.
Where √846 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √846 lies between 29 and 30. 846 is 5 above 841 and 54 below 900, so the root is closer to 29.
- Straight line between 841 and 900: 29.0847 (0% low)
- Tangent from 29, i.e. 29 + 5 ÷ 58: 29.0862 (0% high)
- Tangent from 30, i.e. 30 − 54 ÷ 60: 29.1000 (0.05% high)
For √846 the tangent at 29 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 846 is just 5 above 841.
Finding √846 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 | 846 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 29.1724137931 | 29.0862068966 | 3 |
| 2 | 29.0862068966 | 29.0859513930 | 29.0860791448 | 9 |
| 3 | 29.0860791448 | 29.0860791442 | 29.0860791445 | all 10 shown |
The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √846 = 29.0860791445 to every decimal shown.
√846 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √846 the pattern is [29; 11, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √846 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 |
|---|---|---|
| 29/1 | 29.0000000000 | 8.6 × 10⁻² |
| 320/11 | 29.0909090909 | 4.8 × 10⁻³ |
| 349/12 | 29.0833333333 | 2.7 × 10⁻³ |
| 669/23 | 29.0869565217 | 8.8 × 10⁻⁴ |
| 1,018/35 | 29.0857142857 | 3.6 × 10⁻⁴ |
| 1,687/58 | 29.0862068966 | 1.3 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 846y² = 1. Its smallest solution in positive whole numbers is x = 2,143,295, y = 73,688.
√846 in geometry and everyday measurements
- 846 square feet is 78.6 m². Laid out as a square — a small house footprint or a lot — it is about 29.09 ft (29 ft 1 in) on a side.
- 846 is not a sum of two whole-number squares — the prime factor 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √846 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 29 box, because 1² + 2² + 29² = 846.
- Since √846 = 3√94, a length of √846 is exactly 3 copies of the length √94 laid end to end.
Square roots near √846 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √843 | √843 | 29.0345 | No |
| √844 | 2√211 | 29.0517 | No |
| √845 | 13√5 | 29.0689 | No |
| √846 | 3√94 | 29.0861 | No |
| √847 | 11√7 | 29.1033 | No |
| √848 | 4√53 | 29.1204 | No |
| √849 | √849 | 29.1376 | No |
- The cube root of 846 is about 9.457800.
- Squaring undoes the root: (√846)² = 846, while 846² = 715,716 — the number whose square root is 846.
Frequently asked questions
What is the square root of 846?
The square root of 846 is 3√94 in simplest radical form, which is about 29.0860791445. The negative root, −29.086079, also squares to 846.
Is the square root of 846 rational or irrational?
Irrational. 846 is not a perfect square — it falls between 841 and 900 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √846 be simplified?
Yes. The largest perfect square dividing 846 is 9, so √846 = √9 × √94 = 3√94.
What is √846 rounded to two decimal places?
√846 ≈ 29.09 to two decimal places (29.1 to one, 29.086 to three). Check: 29.09² = 846.2281, close to 846.