√105 at a glance
- Exact value
- √105
- Decimal (10 places)
- 10.2469507660
- Rounded
- 10.2 · 10.25 · 10.247
- Perfect square?
- No — between 10² and 11²
- Rational?
- Irrational
- Both square roots
- ±10.246951
- Prime factorization
- 3 × 5 × 7
- Cube root
- 4.717694
How to simplify √105
The prime factorization of 105 is 3 × 5 × 7. Every prime appears only once, so there is no pair to bring outside the radical — √105 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 105, 3, 5 and 7 appear an odd number of times, so √105 is irrational and 10.2469507660 is a rounded value.
Where √105 sits between perfect squares
100 = 10² and 121 = 11² are the nearest perfect squares, so √105 lies between 10 and 11. 105 is 5 above 100 and 16 below 121, so the root is closer to 10.
- Straight line between 100 and 121: 10.2381 (0.09% low)
- Tangent from 10, i.e. 10 + 5 ÷ 20: 10.2500 (0.03% high)
- Tangent from 11, i.e. 11 − 16 ÷ 22: 10.2727 (0.25% high)
For √105 the tangent at 10 wins, missing by only 0.003. Tangent estimates shine when the number sits close to a perfect square — here 105 is just 5 above 100.
Finding √105 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 105: following the tangent line down to zero simplifies to averaging x with 105 ÷ x.
Start from the nearest whole number, 10 (10² = 100):
| Step | Guess x | 105 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 10.0000000000 | 10.5000000000 | 10.2500000000 | 2 |
| 2 | 10.2500000000 | 10.2439024390 | 10.2469512195 | 6 |
| 3 | 10.2469512195 | 10.2469503124 | 10.2469507660 | 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 √105 = 10.2469507660 to every decimal shown.
√105 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √105 the pattern is [10; 4, 20] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √105 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 | 2.5 × 10⁻¹ |
| 41/4 | 10.2500000000 | 3.0 × 10⁻³ |
| 830/81 | 10.2469135802 | 3.7 × 10⁻⁵ |
| 3,361/328 | 10.2469512195 | 4.5 × 10⁻⁷ |
| 68,050/6,641 | 10.2469507604 | 5.5 × 10⁻⁹ |
| 275,561/26,892 | 10.2469507660 | 6.7 × 10⁻¹¹ |
The same fractions solve Pell’s equation, x² − 105y² = 1. Its smallest solution in positive whole numbers is x = 41, y = 4.
√105 in geometry and everyday measurements
- A square room or garden bed covering 105 square feet measures about 10.25 ft (10 ft 3 in) along each wall.
- 105 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √105 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 10 box, because 1² + 2² + 10² = 105.
Square roots near √105 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √102 | √102 | 10.0995 | No |
| √103 | √103 | 10.1489 | No |
| √104 | 2√26 | 10.1980 | No |
| √105 | √105 | 10.2470 | No |
| √106 | √106 | 10.2956 | No |
| √107 | √107 | 10.3441 | No |
| √108 | 6√3 | 10.3923 | No |
- The cube root of 105 is about 4.717694.
- Four times the radicand doubles the root: √420 = 2 × √105 ≈ 20.493902.
Frequently asked questions
What is the square root of 105?
The square root of 105 is √105, about 10.2469507660. The negative root, −10.246951, also squares to 105.
Is the square root of 105 rational or irrational?
Irrational. 105 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 √105 be simplified?
No. 105 = 3 × 5 × 7 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √105 rounded to two decimal places?
√105 ≈ 10.25 to two decimal places (10.2 to one, 10.247 to three). Check: 10.25² = 105.0625, close to 105.