√108 at a glance
- Exact value
- 6√3
- Decimal (10 places)
- 10.3923048454
- Rounded
- 10.4 · 10.39 · 10.392
- Perfect square?
- No — between 10² and 11²
- Rational?
- Irrational
- Both square roots
- ±10.392305
- Prime factorization
- 2² × 3³
- Cube root
- 4.762203
How to simplify √108
Look for the largest perfect square that divides 108. Here it is 36 (6²), because 108 = 36 × 3 and 3 has no square factor left:
The prime factorization tells the same story: 108 = 2² × 3³. Each pair of equal primes leaves the radical as one factor, so 2 × 3 comes out and 3 stays inside.
108 has 3 square factors (4, 9 and 36). Starting with a smaller one still works but takes more rounds: √108 = 2√27, and √27 can be simplified again. Using 36 straight away finishes in one step.
Check: (6√3)² = 6² × 3 = 36 × 3 = 108. As a decimal, 6√3 = 6 × 1.7320508076 ≈ 10.3923048454.
Where √108 sits between perfect squares
100 = 10² and 121 = 11² are the nearest perfect squares, so √108 lies between 10 and 11. 108 is 8 above 100 and 13 below 121, so the root is closer to 10.
- Straight line between 100 and 121: 10.3810 (0.11% low)
- Tangent from 10, i.e. 10 + 8 ÷ 20: 10.4000 (0.07% high)
- Tangent from 11, i.e. 11 − 13 ÷ 22: 10.4091 (0.16% high)
For √108 the tangent at 10 wins, missing by only 0.0077. Tangent estimates shine when the number sits close to a perfect square — here 108 is just 8 above 100.
Finding √108 with the Babylonian method
Picture a rectangle with an area of 108 and one side x; the other side must be 108 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √108.
Start from the nearest whole number, 10 (10² = 100):
| Step | Guess x | 108 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 10.0000000000 | 10.8000000000 | 10.4000000000 | 2 |
| 2 | 10.4000000000 | 10.3846153846 | 10.3923076923 | 5 |
| 3 | 10.3923076923 | 10.3923019985 | 10.3923048454 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √108 = 10.3923048454 to every decimal shown.
√108 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √108 the pattern is [10; 2, 1, 1, 4, 1, 1, 2, 20] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √108 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 |
|---|---|---|
| 10/1 | 10.0000000000 | 3.9 × 10⁻¹ |
| 21/2 | 10.5000000000 | 1.1 × 10⁻¹ |
| 31/3 | 10.3333333333 | 5.9 × 10⁻² |
| 52/5 | 10.4000000000 | 7.7 × 10⁻³ |
| 239/23 | 10.3913043478 | 1.0 × 10⁻³ |
| 291/28 | 10.3928571429 | 5.5 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 108y² = 1. Its smallest solution in positive whole numbers is x = 1,351, y = 130.
√108 in geometry and everyday measurements
- A square room or garden bed covering 108 square feet measures about 10.39 ft (10 ft 5 in) along each wall.
- 108 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 √108 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 2 × 10 box, because 2² + 2² + 10² = 108.
- Since √108 = 6√3, a length of √108 is exactly 6 copies of the length √3 laid end to end.
Square roots near √108 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √105 | √105 | 10.2470 | No |
| √106 | √106 | 10.2956 | No |
| √107 | √107 | 10.3441 | No |
| √108 | 6√3 | 10.3923 | No |
| √109 | √109 | 10.4403 | No |
| √110 | √110 | 10.4881 | No |
| √111 | √111 | 10.5357 | No |
- The cube root of 108 is about 4.762203.
- Four times the radicand doubles the root: √432 = 2 × √108 ≈ 20.78461.
Frequently asked questions
What is the square root of 108?
The square root of 108 is 6√3 in simplest radical form, which is about 10.3923048454. The negative root, −10.392305, also squares to 108.
Is the square root of 108 rational or irrational?
Irrational. 108 is not a perfect square — it falls between 100 and 121 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √108 be simplified?
Yes. The largest perfect square dividing 108 is 36, so √108 = √36 × √3 = 6√3.
What is √108 rounded to two decimal places?
√108 ≈ 10.39 to two decimal places (10.4 to one, 10.392 to three). Check: 10.39² = 107.9521, close to 108.