√878 at a glance
- Exact value
- √878
- Decimal (10 places)
- 29.6310647801
- Rounded
- 29.6 · 29.63 · 29.631
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.631065
- Prime factorization
- 2 × 439
- Cube root
- 9.575574
How to simplify √878
The prime factorization of 878 is 2 × 439. Every prime appears only once, so there is no pair to bring outside the radical — √878 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 878, 2 and 439 appear an odd number of times, so √878 is irrational and 29.6310647801 is a rounded value.
Where √878 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √878 lies between 29 and 30. 878 is 37 above 841 and 22 below 900, so the root is closer to 30.
- Straight line between 841 and 900: 29.6271 (0.01% low)
- Tangent from 29, i.e. 29 + 37 ÷ 58: 29.6379 (0.02% high)
- Tangent from 30, i.e. 30 − 22 ÷ 60: 29.6333 (0.01% high)
For √878 the tangent at 30 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 878 is just 22 below 900.
Finding √878 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 | 878 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 29.2666666667 | 29.6333333333 | 2 |
| 2 | 29.6333333333 | 29.6287964004 | 29.6310648669 | 7 |
| 3 | 29.6310648669 | 29.6310646932 | 29.6310647801 | 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 √878 = 29.6310647801 to every decimal shown.
√878 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √878 the pattern is [29; 1, 1, 1, 2, 2, 4, 1, 28, 1, 4, 2, 2, …] 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 √878 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 | 6.3 × 10⁻¹ |
| 30/1 | 30.0000000000 | 3.7 × 10⁻¹ |
| 59/2 | 29.5000000000 | 1.3 × 10⁻¹ |
| 89/3 | 29.6666666667 | 3.6 × 10⁻² |
| 237/8 | 29.6250000000 | 6.1 × 10⁻³ |
| 563/19 | 29.6315789474 | 5.1 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 878y² = 1. Its smallest solution in positive whole numbers is x = 9,314,703, y = 314,356.
√878 in geometry and everyday measurements
- 878 square feet is 81.6 m². Laid out as a square — a small house footprint or a lot — it is about 29.63 ft (29 ft 8 in) on a side.
- 878 is not a sum of two whole-number squares — the prime factor 439 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √878 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 6 × 29 box, because 1² + 6² + 29² = 878.
Square roots near √878 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √875 | 5√35 | 29.5804 | No |
| √876 | 2√219 | 29.5973 | No |
| √877 | √877 | 29.6142 | No |
| √878 | √878 | 29.6311 | No |
| √879 | √879 | 29.6479 | No |
| √880 | 4√55 | 29.6648 | No |
| √881 | √881 | 29.6816 | No |
- The cube root of 878 is about 9.575574.
- Squaring undoes the root: (√878)² = 878, while 878² = 770,884 — the number whose square root is 878.
Frequently asked questions
What is the square root of 878?
The square root of 878 is √878, about 29.6310647801. The negative root, −29.631065, also squares to 878.
Is the square root of 878 rational or irrational?
Irrational. 878 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 √878 be simplified?
No. 878 = 2 × 439 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √878 rounded to two decimal places?
√878 ≈ 29.63 to two decimal places (29.6 to one, 29.631 to three). Check: 29.63² = 877.9369, close to 878.