√877 at a glance
- Exact value
- √877
- Decimal (10 places)
- 29.6141857899
- Rounded
- 29.6 · 29.61 · 29.614
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.614186
- Prime factorization
- 877
- Cube root
- 9.571938
How to simplify √877
877 is a prime number, so its only factors are 1 and 877. There is no perfect-square factor to pull out, which means √877 is already in its simplest radical form.
The square root of any prime is irrational. If √877 were a fraction a/b in lowest terms, then a² = 877b², so 877 would divide a — and then 877 would divide b too, contradicting “lowest terms.” That is why the decimal 29.6141857899 is only a rounded value.
Where √877 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √877 lies between 29 and 30. 877 is 36 above 841 and 23 below 900, so the root is closer to 30.
- Straight line between 841 and 900: 29.6102 (0.01% low)
- Tangent from 29, i.e. 29 + 36 ÷ 58: 29.6207 (0.02% high)
- Tangent from 30, i.e. 30 − 23 ÷ 60: 29.6167 (0.01% high)
For √877 the tangent at 30 wins, missing by only 0.0025. Tangent estimates shine when the number sits close to a perfect square — here 877 is just 23 below 900.
Finding √877 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 877: following the tangent line down to zero simplifies to averaging x with 877 ÷ x.
Start from the nearest whole number, 30 (30² = 900):
| Step | Guess x | 877 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 29.2333333333 | 29.6166666667 | 2 |
| 2 | 29.6166666667 | 29.6117051210 | 29.6141858938 | 6 |
| 3 | 29.6141858938 | 29.6141856860 | 29.6141857899 | 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 √877 = 29.6141857899 to every decimal shown.
√877 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √877 the pattern is [29; 1, 1, 1, 1, 2, 4, 1, 1, 4, 2, 1, 1, …] with the block of 15 terms after the semicolon repeating forever (only the first 12 of the 15 are shown). A pattern that never ends is one more proof that √877 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 | 6.1 × 10⁻¹ |
| 30/1 | 30.0000000000 | 3.9 × 10⁻¹ |
| 59/2 | 29.5000000000 | 1.1 × 10⁻¹ |
| 89/3 | 29.6666666667 | 5.2 × 10⁻² |
| 148/5 | 29.6000000000 | 1.4 × 10⁻² |
| 385/13 | 29.6153846154 | 1.2 × 10⁻³ |
The same fractions solve Pell’s equation, x² − 877y² = 1. Its smallest solution in positive whole numbers is x = 116,476,476,553, y = 3,933,131,148. Because the period is odd, the equation with −1 on the right also has a solution: 241,326² − 877 × 8,149² = −1.
√877 in geometry and everyday measurements
- 877 square feet is 81.5 m². Laid out as a square — a small house footprint or a lot — it is about 29.61 ft (29 ft 7 in) on a side.
- 877 = 6² + 29², so by the Pythagorean theorem √877 is the diagonal of a 6 × 29 rectangle — and the distance between the points (0, 0) and (6, 29) on a grid.
Square roots near √877 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √874 | √874 | 29.5635 | No |
| √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 |
- The cube root of 877 is about 9.571938.
- Squaring undoes the root: (√877)² = 877, while 877² = 769,129 — the number whose square root is 877.
Frequently asked questions
What is the square root of 877?
The square root of 877 is √877, about 29.6141857899. The negative root, −29.614186, also squares to 877.
Is the square root of 877 rational or irrational?
Irrational. 877 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 √877 be simplified?
No. 877 is prime, so there is no perfect square to take out of the radical.
What is √877 rounded to two decimal places?
√877 ≈ 29.61 to two decimal places (29.6 to one, 29.614 to three). Check: 29.61² = 876.7521, close to 877.