√918 at a glance
- Exact value
- 3√102
- Decimal (10 places)
- 30.2985148151
- Rounded
- 30.3 · 30.30 · 30.299
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.298515
- Prime factorization
- 2 × 3³ × 17
- Cube root
- 9.718835
How to simplify √918
Look for the largest perfect square that divides 918. Here it is 9 (3²), because 918 = 9 × 102 and 102 has no square factor left:
The prime factorization tells the same story: 918 = 2 × 3³ × 17. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 3 × 17 stays inside.
Check: (3√102)² = 3² × 102 = 9 × 102 = 918. As a decimal, 3√102 = 3 × 10.0995049384 ≈ 30.2985148151.
Where √918 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √918 lies between 30 and 31. 918 is 18 above 900 and 43 below 961, so the root is closer to 30.
- Straight line between 900 and 961: 30.2951 (0.01% low)
- Tangent from 30, i.e. 30 + 18 ÷ 60: 30.3000 (0% high)
- Tangent from 31, i.e. 31 − 43 ÷ 62: 30.3065 (0.03% high)
For √918 the tangent at 30 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 918 is just 18 above 900.
Finding √918 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 | 918 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 30.6000000000 | 30.3000000000 | 2 |
| 2 | 30.3000000000 | 30.2970297030 | 30.2985148515 | 7 |
| 3 | 30.2985148515 | 30.2985147787 | 30.2985148151 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √918 = 30.2985148151 to every decimal shown.
√918 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √918 the pattern is [30; 3, 2, 1, 6, 30, 6, 1, 2, 3, 60] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √918 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 | 3.0 × 10⁻¹ |
| 91/3 | 30.3333333333 | 3.5 × 10⁻² |
| 212/7 | 30.2857142857 | 1.3 × 10⁻² |
| 303/10 | 30.3000000000 | 1.5 × 10⁻³ |
| 2,030/67 | 30.2985074627 | 7.4 × 10⁻⁶ |
| 61,203/2,020 | 30.2985148515 | 3.6 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 918y² = 1. Its smallest solution in positive whole numbers is x = 4,120,901, y = 136,010.
√918 in geometry and everyday measurements
- 918 square feet is 85.3 m². Laid out as a square — a small house footprint or a lot — it is about 30.3 ft (30 ft 4 in) on a side.
- 918 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √918 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 17 × 25 box, because 2² + 17² + 25² = 918.
- Since √918 = 3√102, a length of √918 is exactly 3 copies of the length √102 laid end to end.
Square roots near √918 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √915 | √915 | 30.2490 | No |
| √916 | 2√229 | 30.2655 | No |
| √917 | √917 | 30.2820 | No |
| √918 | 3√102 | 30.2985 | No |
| √919 | √919 | 30.3150 | No |
| √920 | 2√230 | 30.3315 | No |
| √921 | √921 | 30.3480 | No |
- The cube root of 918 is about 9.718835.
- Squaring undoes the root: (√918)² = 918, while 918² = 842,724 — the number whose square root is 918.
Frequently asked questions
What is the square root of 918?
The square root of 918 is 3√102 in simplest radical form, which is about 30.2985148151. The negative root, −30.298515, also squares to 918.
Is the square root of 918 rational or irrational?
Irrational. 918 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 √918 be simplified?
Yes. The largest perfect square dividing 918 is 9, so √918 = √9 × √102 = 3√102.
What is √918 rounded to two decimal places?
√918 ≈ 30.30 to two decimal places (30.3 to one, 30.299 to three). Check: 30.30² = 918.09, close to 918.