√930 at a glance
- Exact value
- √930
- Decimal (10 places)
- 30.4959013640
- Rounded
- 30.5 · 30.50 · 30.496
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.495901
- Prime factorization
- 2 × 3 × 5 × 31
- Cube root
- 9.761000
How to simplify √930
The prime factorization of 930 is 2 × 3 × 5 × 31. Every prime appears only once, so there is no pair to bring outside the radical — √930 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 930, 2, 3, 5 and 31 appear an odd number of times, so √930 is irrational and 30.4959013640 is a rounded value.
Where √930 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √930 lies between 30 and 31. 930 is 30 above 900 and 31 below 961, so the root is closer to 30.
- Straight line between 900 and 961: 30.4918 (0.01% low)
- Tangent from 30, i.e. 30 + 30 ÷ 60: 30.5000 (0.01% high)
- Tangent from 31, i.e. 31 − 31 ÷ 62: 30.5000 (0.01% high)
For √930 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.
Finding √930 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 | 930 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 31.0000000000 | 30.5000000000 | 2 |
| 2 | 30.5000000000 | 30.4918032787 | 30.4959016393 | 6 |
| 3 | 30.4959016393 | 30.4959010886 | 30.4959013640 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √930 = 30.4959013640 to every decimal shown.
√930 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √930 the pattern is [30; 2, 60] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √930 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 |
|---|---|---|
| 30/1 | 30.0000000000 | 5.0 × 10⁻¹ |
| 61/2 | 30.5000000000 | 4.1 × 10⁻³ |
| 3,690/121 | 30.4958677686 | 3.4 × 10⁻⁵ |
| 7,441/244 | 30.4959016393 | 2.8 × 10⁻⁷ |
| 450,150/14,761 | 30.4959013617 | 2.3 × 10⁻⁹ |
| 907,741/29,766 | 30.4959013640 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 930y² = 1. Its smallest solution in positive whole numbers is x = 61, y = 2.
√930 in geometry and everyday measurements
- 930 square feet is 86.4 m². Laid out as a square — a small house footprint or a lot — it is about 30.5 ft (30 ft 6 in) on a side.
- 930 is not a sum of two whole-number squares — the prime factor 3 and 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √930 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 20 × 23 box, because 1² + 20² + 23² = 930.
Square roots near √930 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √927 | 3√103 | 30.4467 | No |
| √928 | 4√58 | 30.4631 | No |
| √929 | √929 | 30.4795 | No |
| √930 | √930 | 30.4959 | No |
| √931 | 7√19 | 30.5123 | No |
| √932 | 2√233 | 30.5287 | No |
| √933 | √933 | 30.5450 | No |
- The cube root of 930 is about 9.761000.
- Squaring undoes the root: (√930)² = 930, while 930² = 864,900 — the number whose square root is 930.
Frequently asked questions
What is the square root of 930?
The square root of 930 is √930, about 30.4959013640. The negative root, −30.495901, also squares to 930.
Is the square root of 930 rational or irrational?
Irrational. 930 is not a perfect square — it falls between 900 and 961 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √930 be simplified?
No. 930 = 2 × 3 × 5 × 31 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √930 rounded to two decimal places?
√930 ≈ 30.50 to two decimal places (30.5 to one, 30.496 to three). Check: 30.50² = 930.25, close to 930.