√897 at a glance
- Exact value
- √897
- Decimal (10 places)
- 29.9499582637
- Rounded
- 29.9 · 29.95 · 29.950
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.949958
- Prime factorization
- 3 × 13 × 23
- Cube root
- 9.644154
How to simplify √897
The prime factorization of 897 is 3 × 13 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √897 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 897, 3, 13 and 23 appear an odd number of times, so √897 is irrational and 29.9499582637 is a rounded value.
Where √897 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √897 lies between 29 and 30. 897 is 56 above 841 and 3 below 900, so the root is closer to 30.
- Straight line between 841 and 900: 29.9492 (0% low)
- Tangent from 29, i.e. 29 + 56 ÷ 58: 29.9655 (0.05% high)
- Tangent from 30, i.e. 30 − 3 ÷ 60: 29.9500 (0% high)
For √897 the tangent at 30 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 897 is just 3 below 900.
Finding √897 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 897: following the tangent line down to zero simplifies to averaging x with 897 ÷ x.
Start from the nearest whole number, 30 (30² = 900):
| Step | Guess x | 897 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 29.9000000000 | 29.9500000000 | 4 |
| 2 | 29.9500000000 | 29.9499165275 | 29.9499582638 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √897 = 29.9499582637 to every decimal shown.
√897 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √897 the pattern is [29; 1, 18, 1, 58] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √897 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 | 9.5 × 10⁻¹ |
| 30/1 | 30.0000000000 | 5.0 × 10⁻² |
| 569/19 | 29.9473684211 | 2.6 × 10⁻³ |
| 599/20 | 29.9500000000 | 4.2 × 10⁻⁵ |
| 35,311/1,179 | 29.9499575912 | 6.7 × 10⁻⁷ |
| 35,910/1,199 | 29.9499582986 | 3.5 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 897y² = 1. Its smallest solution in positive whole numbers is x = 599, y = 20.
√897 in geometry and everyday measurements
- 897 square feet is 83.3 m². Laid out as a square — a small house footprint or a lot — it is about 29.95 ft (29 ft 11 in) on a side.
- 897 is not a sum of two whole-number squares — the prime factor 3 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √897 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 16 × 25 box, because 4² + 16² + 25² = 897.
Square roots near √897 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √894 | √894 | 29.8998 | No |
| √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 |
- The cube root of 897 is about 9.644154.
- Squaring undoes the root: (√897)² = 897, while 897² = 804,609 — the number whose square root is 897.
Frequently asked questions
What is the square root of 897?
The square root of 897 is √897, about 29.9499582637. The negative root, −29.949958, also squares to 897.
Is the square root of 897 rational or irrational?
Irrational. 897 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 √897 be simplified?
No. 897 = 3 × 13 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √897 rounded to two decimal places?
√897 ≈ 29.95 to two decimal places (29.9 to one, 29.950 to three). Check: 29.95² = 897.0025, close to 897.