√898 at a glance
- Exact value
- √898
- Decimal (10 places)
- 29.9666481275
- Rounded
- 30.0 · 29.97 · 29.967
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.966648
- Prime factorization
- 2 × 449
- Cube root
- 9.647737
How to simplify √898
The prime factorization of 898 is 2 × 449. Every prime appears only once, so there is no pair to bring outside the radical — √898 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 898, 2 and 449 appear an odd number of times, so √898 is irrational and 29.9666481275 is a rounded value.
Where √898 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √898 lies between 29 and 30. 898 is 57 above 841 and 2 below 900, so the root is closer to 30.
- Straight line between 841 and 900: 29.9661 (0% low)
- Tangent from 29, i.e. 29 + 57 ÷ 58: 29.9828 (0.05% high)
- Tangent from 30, i.e. 30 − 2 ÷ 60: 29.9667 (0% high)
For √898 the tangent at 30 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 898 is just 2 below 900.
Finding √898 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 | 898 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 29.9333333333 | 29.9666666667 | 4 |
| 2 | 29.9666666667 | 29.9666295884 | 29.9666481275 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √898 = 29.9666481275 to every decimal shown.
√898 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √898 the pattern is [29; 1, 28, 1, 58] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √898 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 | 9.7 × 10⁻¹ |
| 30/1 | 30.0000000000 | 3.3 × 10⁻² |
| 869/29 | 29.9655172414 | 1.1 × 10⁻³ |
| 899/30 | 29.9666666667 | 1.9 × 10⁻⁵ |
| 53,011/1,769 | 29.9666478236 | 3.0 × 10⁻⁷ |
| 53,910/1,799 | 29.9666481379 | 1.0 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 898y² = 1. Its smallest solution in positive whole numbers is x = 899, y = 30.
√898 in geometry and everyday measurements
- 898 square feet is 83.4 m². Laid out as a square — a small house footprint or a lot — it is about 29.97 ft (30 ft) on a side.
- 898 = 13² + 27², so by the Pythagorean theorem √898 is the diagonal of a 13 × 27 rectangle — and the distance between the points (0, 0) and (13, 27) on a grid.
Square roots near √898 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √895 | √895 | 29.9166 | No |
| √896 | 8√14 | 29.9333 | No |
| √897 | √897 | 29.9500 | No |
| √898 | √898 | 29.9666 | No |
| √899 | √899 | 29.9833 | No |
| √900 | 30 | 30.0000 | Yes |
| √901 | √901 | 30.0167 | No |
- The cube root of 898 is about 9.647737.
- Squaring undoes the root: (√898)² = 898, while 898² = 806,404 — the number whose square root is 898.
Frequently asked questions
What is the square root of 898?
The square root of 898 is √898, about 29.9666481275. The negative root, −29.966648, also squares to 898.
Is the square root of 898 rational or irrational?
Irrational. 898 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 √898 be simplified?
No. 898 = 2 × 449 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √898 rounded to two decimal places?
√898 ≈ 29.97 to two decimal places (30.0 to one, 29.967 to three). Check: 29.97² = 898.2009, close to 898.