√910 at a glance
- Exact value
- √910
- Decimal (10 places)
- 30.1662062580
- Rounded
- 30.2 · 30.17 · 30.166
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.166206
- Prime factorization
- 2 × 5 × 7 × 13
- Cube root
- 9.690521
How to simplify √910
The prime factorization of 910 is 2 × 5 × 7 × 13. Every prime appears only once, so there is no pair to bring outside the radical — √910 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 910, 2, 5, 7 and 13 appear an odd number of times, so √910 is irrational and 30.1662062580 is a rounded value.
Where √910 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √910 lies between 30 and 31. 910 is 10 above 900 and 51 below 961, so the root is closer to 30.
- Straight line between 900 and 961: 30.1639 (0.01% low)
- Tangent from 30, i.e. 30 + 10 ÷ 60: 30.1667 (0% high)
- Tangent from 31, i.e. 31 − 51 ÷ 62: 30.1774 (0.04% high)
For √910 the tangent at 30 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 910 is just 10 above 900.
Finding √910 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 | 910 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 30.3333333333 | 30.1666666667 | 3 |
| 2 | 30.1666666667 | 30.1657458564 | 30.1662062615 | 8 |
| 3 | 30.1662062615 | 30.1662062545 | 30.1662062580 | 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 √910 = 30.1662062580 to every decimal shown.
√910 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √910 the pattern is [30; 6, 60] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √910 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 | 1.7 × 10⁻¹ |
| 181/6 | 30.1666666667 | 4.6 × 10⁻⁴ |
| 10,890/361 | 30.1662049861 | 1.3 × 10⁻⁶ |
| 65,521/2,172 | 30.1662062615 | 3.5 × 10⁻⁹ |
| 3,942,150/130,681 | 30.1662062580 | < 10⁻¹⁰ |
| 23,718,421/786,258 | 30.1662062580 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 910y² = 1. Its smallest solution in positive whole numbers is x = 181, y = 6.
√910 in geometry and everyday measurements
- 910 square feet is 84.5 m². Laid out as a square — a small house footprint or a lot — it is about 30.17 ft (30 ft 2 in) on a side.
- 910 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √910 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 30 box, because 1² + 3² + 30² = 910.
Square roots near √910 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √907 | √907 | 30.1164 | No |
| √908 | 2√227 | 30.1330 | No |
| √909 | 3√101 | 30.1496 | No |
| √910 | √910 | 30.1662 | No |
| √911 | √911 | 30.1828 | No |
| √912 | 4√57 | 30.1993 | No |
| √913 | √913 | 30.2159 | No |
- The cube root of 910 is about 9.690521.
- Squaring undoes the root: (√910)² = 910, while 910² = 828,100 — the number whose square root is 910.
Frequently asked questions
What is the square root of 910?
The square root of 910 is √910, about 30.1662062580. The negative root, −30.166206, also squares to 910.
Is the square root of 910 rational or irrational?
Irrational. 910 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 √910 be simplified?
No. 910 = 2 × 5 × 7 × 13 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √910 rounded to two decimal places?
√910 ≈ 30.17 to two decimal places (30.2 to one, 30.166 to three). Check: 30.17² = 910.2289, close to 910.