√882 at a glance
- Exact value
- 21√2
- Decimal (10 places)
- 29.6984848098
- Rounded
- 29.7 · 29.70 · 29.698
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.698485
- Prime factorization
- 2 × 3² × 7²
- Cube root
- 9.590094
How to simplify √882
Look for the largest perfect square that divides 882. Here it is 441 (21²), because 882 = 441 × 2 and 2 has no square factor left:
The prime factorization tells the same story: 882 = 2 × 3² × 7². Each pair of equal primes leaves the radical as one factor, so 3 × 7 comes out and 2 stays inside.
882 has 3 square factors (9, 49 and 441). Starting with a smaller one still works but takes more rounds: √882 = 3√98, and √98 can be simplified again. Using 441 straight away finishes in one step.
Check: (21√2)² = 21² × 2 = 441 × 2 = 882. As a decimal, 21√2 = 21 × 1.4142135624 ≈ 29.6984848098.
Where √882 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √882 lies between 29 and 30. 882 is 41 above 841 and 18 below 900, so the root is closer to 30.
- Straight line between 841 and 900: 29.6949 (0.01% low)
- Tangent from 29, i.e. 29 + 41 ÷ 58: 29.7069 (0.03% high)
- Tangent from 30, i.e. 30 − 18 ÷ 60: 29.7000 (0.01% high)
For √882 the tangent at 30 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 882 is just 18 below 900.
Finding √882 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 | 882 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 29.4000000000 | 29.7000000000 | 2 |
| 2 | 29.7000000000 | 29.6969696970 | 29.6984848485 | 7 |
| 3 | 29.6984848485 | 29.6984847712 | 29.6984848098 | 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 √882 = 29.6984848098 to every decimal shown.
√882 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √882 the pattern is [29; 1, 2, 3, 6, 3, 2, 1, 58] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √882 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 | 7.0 × 10⁻¹ |
| 30/1 | 30.0000000000 | 3.0 × 10⁻¹ |
| 89/3 | 29.6666666667 | 3.2 × 10⁻² |
| 297/10 | 29.7000000000 | 1.5 × 10⁻³ |
| 1,871/63 | 29.6984126984 | 7.2 × 10⁻⁵ |
| 5,910/199 | 29.6984924623 | 7.7 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 882y² = 1. Its smallest solution in positive whole numbers is x = 19,601, y = 660.
√882 in geometry and everyday measurements
- 882 square feet is 81.9 m². Laid out as a square — a small house footprint or a lot — it is about 29.7 ft (29 ft 8 in) on a side.
- 882 = 21² + 21², so by the Pythagorean theorem √882 is the diagonal of a 21 × 21 rectangle — and the distance between the points (0, 0) and (21, 21) on a grid.
- Since √882 = 21√2, a length of √882 is exactly 21 copies of the length √2 laid end to end.
Square roots near √882 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √879 | √879 | 29.6479 | No |
| √880 | 4√55 | 29.6648 | No |
| √881 | √881 | 29.6816 | No |
| √882 | 21√2 | 29.6985 | No |
| √883 | √883 | 29.7153 | No |
| √884 | 2√221 | 29.7321 | No |
| √885 | √885 | 29.7489 | No |
- The cube root of 882 is about 9.590094.
- Squaring undoes the root: (√882)² = 882, while 882² = 777,924 — the number whose square root is 882.
Frequently asked questions
What is the square root of 882?
The square root of 882 is 21√2 in simplest radical form, which is about 29.6984848098. The negative root, −29.698485, also squares to 882.
Is the square root of 882 rational or irrational?
Irrational. 882 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 √882 be simplified?
Yes. The largest perfect square dividing 882 is 441, so √882 = √441 × √2 = 21√2.
What is √882 rounded to two decimal places?
√882 ≈ 29.70 to two decimal places (29.7 to one, 29.698 to three). Check: 29.70² = 882.09, close to 882.