√874 at a glance
- Exact value
- √874
- Decimal (10 places)
- 29.5634909982
- Rounded
- 29.6 · 29.56 · 29.563
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.563491
- Prime factorization
- 2 × 19 × 23
- Cube root
- 9.561011
How to simplify √874
The prime factorization of 874 is 2 × 19 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √874 is already in simplest form (a "square-free" radicand).
A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 874, 2, 19 and 23 appear an odd number of times, so √874 is irrational and 29.5634909982 is a rounded value.
Where √874 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √874 lies between 29 and 30. 874 is 33 above 841 and 26 below 900, so the root is closer to 30.
- Straight line between 841 and 900: 29.5593 (0.01% low)
- Tangent from 29, i.e. 29 + 33 ÷ 58: 29.5690 (0.02% high)
- Tangent from 30, i.e. 30 − 26 ÷ 60: 29.5667 (0.01% high)
For √874 the tangent at 30 wins, missing by only 0.0032. Tangent estimates shine when the number sits close to a perfect square — here 874 is just 26 below 900.
Finding √874 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, 30 (30² = 900):
| Step | Guess x | 874 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 29.1333333333 | 29.5666666667 | 2 |
| 2 | 29.5666666667 | 29.5603156708 | 29.5634911687 | 6 |
| 3 | 29.5634911687 | 29.5634908276 | 29.5634909982 | 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 √874 = 29.5634909982 to every decimal shown.
√874 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √874 the pattern is [29; 1, 1, 3, 2, 3, 1, 1, 58] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √874 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 | 5.6 × 10⁻¹ |
| 30/1 | 30.0000000000 | 4.4 × 10⁻¹ |
| 59/2 | 29.5000000000 | 6.3 × 10⁻² |
| 207/7 | 29.5714285714 | 7.9 × 10⁻³ |
| 473/16 | 29.5625000000 | 9.9 × 10⁻⁴ |
| 1,626/55 | 29.5636363636 | 1.5 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 874y² = 1. Its smallest solution in positive whole numbers is x = 3,725, y = 126.
√874 in geometry and everyday measurements
- 874 square feet is 81.2 m². Laid out as a square — a small house footprint or a lot — it is about 29.56 ft (29 ft 7 in) on a side.
- 874 is not a sum of two whole-number squares — the prime factor 19 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √874 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 12 × 27 box, because 1² + 12² + 27² = 874.
Square roots near √874 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √871 | √871 | 29.5127 | No |
| √872 | 2√218 | 29.5296 | No |
| √873 | 3√97 | 29.5466 | No |
| √874 | √874 | 29.5635 | No |
| √875 | 5√35 | 29.5804 | No |
| √876 | 2√219 | 29.5973 | No |
| √877 | √877 | 29.6142 | No |
- The cube root of 874 is about 9.561011.
- Squaring undoes the root: (√874)² = 874, while 874² = 763,876 — the number whose square root is 874.
Frequently asked questions
What is the square root of 874?
The square root of 874 is √874, about 29.5634909982. The negative root, −29.563491, also squares to 874.
Is the square root of 874 rational or irrational?
Irrational. 874 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 √874 be simplified?
No. 874 = 2 × 19 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √874 rounded to two decimal places?
√874 ≈ 29.56 to two decimal places (29.6 to one, 29.563 to three). Check: 29.56² = 873.7936, close to 874.