√893 at a glance
- Exact value
- √893
- Decimal (10 places)
- 29.8831055950
- Rounded
- 29.9 · 29.88 · 29.883
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.883106
- Prime factorization
- 19 × 47
- Cube root
- 9.629797
How to simplify √893
The prime factorization of 893 is 19 × 47. Every prime appears only once, so there is no pair to bring outside the radical — √893 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 893, 19 and 47 appear an odd number of times, so √893 is irrational and 29.8831055950 is a rounded value.
Where √893 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √893 lies between 29 and 30. 893 is 52 above 841 and 7 below 900, so the root is closer to 30.
- Straight line between 841 and 900: 29.8814 (0.01% low)
- Tangent from 29, i.e. 29 + 52 ÷ 58: 29.8966 (0.04% high)
- Tangent from 30, i.e. 30 − 7 ÷ 60: 29.8833 (0% high)
For √893 the tangent at 30 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 893 is just 7 below 900.
Finding √893 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 893: following the tangent line down to zero simplifies to averaging x with 893 ÷ x.
Start from the nearest whole number, 30 (30² = 900):
| Step | Guess x | 893 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 29.7666666667 | 29.8833333333 | 3 |
| 2 | 29.8833333333 | 29.8828778583 | 29.8831055958 | 9 |
| 3 | 29.8831055958 | 29.8831055941 | 29.8831055950 | all 10 shown |
The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √893 = 29.8831055950 to every decimal shown.
√893 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √893 the pattern is [29; 1, 7, 1, 1, 4, 14, 1, 2, 1, 1, 2, 1, …] with the block of 22 terms after the semicolon repeating forever (only the first 12 of the 22 are shown). A pattern that never ends is one more proof that √893 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 | 8.8 × 10⁻¹ |
| 30/1 | 30.0000000000 | 1.2 × 10⁻¹ |
| 239/8 | 29.8750000000 | 8.1 × 10⁻³ |
| 269/9 | 29.8888888889 | 5.8 × 10⁻³ |
| 508/17 | 29.8823529412 | 7.5 × 10⁻⁴ |
| 2,301/77 | 29.8831168831 | 1.1 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 893y² = 1. Its smallest solution in positive whole numbers is x = 6,091,434,999, y = 203,842,100.
√893 in geometry and everyday measurements
- 893 square feet is 83 m². Laid out as a square — a small house footprint or a lot — it is about 29.88 ft (29 ft 11 in) on a side.
- 893 is not a sum of two whole-number squares — the prime factor 19 and 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √893 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 10 × 28 box, because 3² + 10² + 28² = 893.
Square roots near √893 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √890 | √890 | 29.8329 | No |
| √891 | 9√11 | 29.8496 | No |
| √892 | 2√223 | 29.8664 | No |
| √893 | √893 | 29.8831 | No |
| √894 | √894 | 29.8998 | No |
| √895 | √895 | 29.9166 | No |
| √896 | 8√14 | 29.9333 | No |
- The cube root of 893 is about 9.629797.
- Squaring undoes the root: (√893)² = 893, while 893² = 797,449 — the number whose square root is 893.
Frequently asked questions
What is the square root of 893?
The square root of 893 is √893, about 29.8831055950. The negative root, −29.883106, also squares to 893.
Is the square root of 893 rational or irrational?
Irrational. 893 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 √893 be simplified?
No. 893 = 19 × 47 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √893 rounded to two decimal places?
√893 ≈ 29.88 to two decimal places (29.9 to one, 29.883 to three). Check: 29.88² = 892.8144, close to 893.