√879 at a glance
- Exact value
- √879
- Decimal (10 places)
- 29.6479341607
- Rounded
- 29.6 · 29.65 · 29.648
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.647934
- Prime factorization
- 3 × 293
- Cube root
- 9.579208
How to simplify √879
The prime factorization of 879 is 3 × 293. Every prime appears only once, so there is no pair to bring outside the radical — √879 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 879, 3 and 293 appear an odd number of times, so √879 is irrational and 29.6479341607 is a rounded value.
Where √879 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √879 lies between 29 and 30. 879 is 38 above 841 and 21 below 900, so the root is closer to 30.
- Straight line between 841 and 900: 29.6441 (0.01% low)
- Tangent from 29, i.e. 29 + 38 ÷ 58: 29.6552 (0.02% high)
- Tangent from 30, i.e. 30 − 21 ÷ 60: 29.6500 (0.01% high)
For √879 the tangent at 30 wins, missing by only 0.0021. Tangent estimates shine when the number sits close to a perfect square — here 879 is just 21 below 900.
Finding √879 with the Babylonian method
If a guess is too big, 879 ÷ 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√879) in one step.
Start from the nearest whole number, 30 (30² = 900):
| Step | Guess x | 879 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 29.3000000000 | 29.6500000000 | 2 |
| 2 | 29.6500000000 | 29.6458684654 | 29.6479342327 | 7 |
| 3 | 29.6479342327 | 29.6479340888 | 29.6479341607 | 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 √879 = 29.6479341607 to every decimal shown.
√879 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √879 the pattern is [29; 1, 1, 1, 5, 3, 1, 3, 2, 9, 2, 3, 1, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √879 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 | 6.5 × 10⁻¹ |
| 30/1 | 30.0000000000 | 3.5 × 10⁻¹ |
| 59/2 | 29.5000000000 | 1.5 × 10⁻¹ |
| 89/3 | 29.6666666667 | 1.9 × 10⁻² |
| 504/17 | 29.6470588235 | 8.8 × 10⁻⁴ |
| 1,601/54 | 29.6481481481 | 2.1 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 879y² = 1. Its smallest solution in positive whole numbers is x = 107,245,324, y = 3,617,295.
√879 in geometry and everyday measurements
- 879 square feet is 81.7 m². Laid out as a square — a small house footprint or a lot — it is about 29.65 ft (29 ft 8 in) on a side.
- 879 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 √879 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √879 as its space diagonal.
Square roots near √879 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √882 | 21√2 | 29.6985 | No |
- The cube root of 879 is about 9.579208.
- Squaring undoes the root: (√879)² = 879, while 879² = 772,641 — the number whose square root is 879.
Frequently asked questions
What is the square root of 879?
The square root of 879 is √879, about 29.6479341607. The negative root, −29.647934, also squares to 879.
Is the square root of 879 rational or irrational?
Irrational. 879 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 √879 be simplified?
No. 879 = 3 × 293 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √879 rounded to two decimal places?
√879 ≈ 29.65 to two decimal places (29.6 to one, 29.648 to three). Check: 29.65² = 879.1225, close to 879.