√889 at a glance
- Exact value
- √889
- Decimal (10 places)
- 29.8161030318
- Rounded
- 29.8 · 29.82 · 29.816
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.816103
- Prime factorization
- 7 × 127
- Cube root
- 9.615398
How to simplify √889
The prime factorization of 889 is 7 × 127. Every prime appears only once, so there is no pair to bring outside the radical — √889 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 889, 7 and 127 appear an odd number of times, so √889 is irrational and 29.8161030318 is a rounded value.
Where √889 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √889 lies between 29 and 30. 889 is 48 above 841 and 11 below 900, so the root is closer to 30.
- Straight line between 841 and 900: 29.8136 (0.01% low)
- Tangent from 29, i.e. 29 + 48 ÷ 58: 29.8276 (0.04% high)
- Tangent from 30, i.e. 30 − 11 ÷ 60: 29.8167 (0% high)
For √889 the tangent at 30 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 889 is just 11 below 900.
Finding √889 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 889: following the tangent line down to zero simplifies to averaging x with 889 ÷ x.
Start from the nearest whole number, 30 (30² = 900):
| Step | Guess x | 889 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 29.6333333333 | 29.8166666667 | 3 |
| 2 | 29.8166666667 | 29.8155394075 | 29.8161030371 | 8 |
| 3 | 29.8161030371 | 29.8161030264 | 29.8161030318 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √889 = 29.8161030318 to every decimal shown.
√889 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √889 the pattern is [29; 1, 4, 2, 3, 1, 1, 11, 2, 1, 3, 19, 1, …] with the block of 42 terms after the semicolon repeating forever (only the first 12 of the 42 are shown). A pattern that never ends is one more proof that √889 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 | 8.2 × 10⁻¹ |
| 30/1 | 30.0000000000 | 1.8 × 10⁻¹ |
| 149/5 | 29.8000000000 | 1.6 × 10⁻² |
| 328/11 | 29.8181818182 | 2.1 × 10⁻³ |
| 1,133/38 | 29.8157894737 | 3.1 × 10⁻⁴ |
| 1,461/49 | 29.8163265306 | 2.2 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 889y² = 1. Its smallest solution in positive whole numbers is x = 13,231,974,717,803,657,215, y = 443,786,188,413,453,504 — 20 digits for x, even though 889 is small, which is what makes Pell’s equation famous.
√889 in geometry and everyday measurements
- 889 square feet is 82.6 m². Laid out as a square — a small house footprint or a lot — it is about 29.82 ft (29 ft 10 in) on a side.
- 889 is not a sum of two whole-number squares — the prime factor 7 and 127 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √889 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 12 × 27 box, because 4² + 12² + 27² = 889.
Square roots near √889 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √886 | √886 | 29.7658 | No |
| √887 | √887 | 29.7825 | No |
| √888 | 2√222 | 29.7993 | No |
| √889 | √889 | 29.8161 | No |
| √890 | √890 | 29.8329 | No |
| √891 | 9√11 | 29.8496 | No |
| √892 | 2√223 | 29.8664 | No |
- The cube root of 889 is about 9.615398.
- Squaring undoes the root: (√889)² = 889, while 889² = 790,321 — the number whose square root is 889.
Frequently asked questions
What is the square root of 889?
The square root of 889 is √889, about 29.8161030318. The negative root, −29.816103, also squares to 889.
Is the square root of 889 rational or irrational?
Irrational. 889 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 √889 be simplified?
No. 889 = 7 × 127 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √889 rounded to two decimal places?
√889 ≈ 29.82 to two decimal places (29.8 to one, 29.816 to three). Check: 29.82² = 889.2324, close to 889.