√867 at a glance
- Exact value
- 17√3
- Decimal (10 places)
- 29.4448637287
- Rounded
- 29.4 · 29.44 · 29.445
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.444864
- Prime factorization
- 3 × 17²
- Cube root
- 9.535417
How to simplify √867
Look for the largest perfect square that divides 867. Here it is 289 (17²), because 867 = 289 × 3 and 3 has no square factor left:
The prime factorization tells the same story: 867 = 3 × 17². Each pair of equal primes leaves the radical as one factor, so 17 comes out and 3 stays inside.
Check: (17√3)² = 17² × 3 = 289 × 3 = 867. As a decimal, 17√3 = 17 × 1.7320508076 ≈ 29.4448637287.
Where √867 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √867 lies between 29 and 30. 867 is 26 above 841 and 33 below 900, so the root is closer to 29.
- Straight line between 841 and 900: 29.4407 (0.01% low)
- Tangent from 29, i.e. 29 + 26 ÷ 58: 29.4483 (0.01% high)
- Tangent from 30, i.e. 30 − 33 ÷ 60: 29.4500 (0.02% high)
For √867 the tangent at 29 wins, missing by only 0.0034. Tangent estimates shine when the number sits close to a perfect square — here 867 is just 26 above 841.
Finding √867 with the Babylonian method
If a guess is too big, 867 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√867) in one step.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 867 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 29.8965517241 | 29.4482758621 | 2 |
| 2 | 29.4482758621 | 29.4414519906 | 29.4448639264 | 6 |
| 3 | 29.4448639264 | 29.4448635310 | 29.4448637287 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √867 = 29.4448637287 to every decimal shown.
√867 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √867 the pattern is [29; 2, 4, 29, 4, 2, 58] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √867 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 | 4.4 × 10⁻¹ |
| 59/2 | 29.5000000000 | 5.5 × 10⁻² |
| 265/9 | 29.4444444444 | 4.2 × 10⁻⁴ |
| 7,744/263 | 29.4448669202 | 3.2 × 10⁻⁶ |
| 31,241/1,061 | 29.4448633365 | 3.9 × 10⁻⁷ |
| 70,226/2,385 | 29.4448637317 | 3.0 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 867y² = 1. Its smallest solution in positive whole numbers is x = 70,226, y = 2,385.
√867 in geometry and everyday measurements
- 867 square feet is 80.5 m². Laid out as a square — a small house footprint or a lot — it is about 29.44 ft (29 ft 5 in) on a side.
- 867 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √867 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 29 box, because 1² + 5² + 29² = 867.
- Since √867 = 17√3, a length of √867 is exactly 17 copies of the length √3 laid end to end.
Square roots near √867 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √864 | 12√6 | 29.3939 | No |
| √865 | √865 | 29.4109 | No |
| √866 | √866 | 29.4279 | No |
| √867 | 17√3 | 29.4449 | No |
| √868 | 2√217 | 29.4618 | No |
| √869 | √869 | 29.4788 | No |
| √870 | √870 | 29.4958 | No |
- The cube root of 867 is about 9.535417.
- Squaring undoes the root: (√867)² = 867, while 867² = 751,689 — the number whose square root is 867.
Frequently asked questions
What is the square root of 867?
The square root of 867 is 17√3 in simplest radical form, which is about 29.4448637287. The negative root, −29.444864, also squares to 867.
Is the square root of 867 rational or irrational?
Irrational. 867 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 √867 be simplified?
Yes. The largest perfect square dividing 867 is 289, so √867 = √289 × √3 = 17√3.
What is √867 rounded to two decimal places?
√867 ≈ 29.44 to two decimal places (29.4 to one, 29.445 to three). Check: 29.44² = 866.7136, close to 867.