√107 at a glance
- Exact value
- √107
- Decimal (10 places)
- 10.3440804328
- Rounded
- 10.3 · 10.34 · 10.344
- Perfect square?
- No — between 10² and 11²
- Rational?
- Irrational
- Both square roots
- ±10.344080
- Prime factorization
- 107
- Cube root
- 4.747459
How to simplify √107
107 is a prime number, so its only factors are 1 and 107. There is no perfect-square factor to pull out, which means √107 is already in its simplest radical form.
The square root of any prime is irrational. If √107 were a fraction a/b in lowest terms, then a² = 107b², so 107 would divide a — and then 107 would divide b too, contradicting “lowest terms.” That is why the decimal 10.3440804328 is only a rounded value.
Where √107 sits between perfect squares
100 = 10² and 121 = 11² are the nearest perfect squares, so √107 lies between 10 and 11. 107 is 7 above 100 and 14 below 121, so the root is closer to 10.
- Straight line between 100 and 121: 10.3333 (0.1% low)
- Tangent from 10, i.e. 10 + 7 ÷ 20: 10.3500 (0.06% high)
- Tangent from 11, i.e. 11 − 14 ÷ 22: 10.3636 (0.19% high)
For √107 the tangent at 10 wins, missing by only 0.0059. Tangent estimates shine when the number sits close to a perfect square — here 107 is just 7 above 100.
Finding √107 with the Babylonian method
If a guess is too big, 107 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√107) in one step.
Start from the nearest whole number, 10 (10² = 100):
| Step | Guess x | 107 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 10.0000000000 | 10.7000000000 | 10.3500000000 | 2 |
| 2 | 10.3500000000 | 10.3381642512 | 10.3440821256 | 5 |
| 3 | 10.3440821256 | 10.3440787400 | 10.3440804328 | 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 √107 = 10.3440804328 to every decimal shown.
√107 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √107 the pattern is [10; 2, 1, 9, 1, 2, 20] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √107 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 | 3.4 × 10⁻¹ |
| 21/2 | 10.5000000000 | 1.6 × 10⁻¹ |
| 31/3 | 10.3333333333 | 1.1 × 10⁻² |
| 300/29 | 10.3448275862 | 7.5 × 10⁻⁴ |
| 331/32 | 10.3437500000 | 3.3 × 10⁻⁴ |
| 962/93 | 10.3440860215 | 5.6 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 107y² = 1. Its smallest solution in positive whole numbers is x = 962, y = 93.
√107 in geometry and everyday measurements
- A square room or garden bed covering 107 square feet measures about 10.34 ft (10 ft 4 in) along each wall.
- 107 is not a sum of two whole-number squares — 107 is itself a prime that is one less than a multiple of 4, which rules that out — so √107 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 9 box, because 1² + 5² + 9² = 107.
Square roots near √107 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √109 | √109 | 10.4403 | No |
| √110 | √110 | 10.4881 | No |
- The cube root of 107 is about 4.747459.
- Four times the radicand doubles the root: √428 = 2 × √107 ≈ 20.688161.
Frequently asked questions
What is the square root of 107?
The square root of 107 is √107, about 10.3440804328. The negative root, −10.344080, also squares to 107.
Is the square root of 107 rational or irrational?
Irrational. 107 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 √107 be simplified?
No. 107 is prime, so there is no perfect square to take out of the radical.
What is √107 rounded to two decimal places?
√107 ≈ 10.34 to two decimal places (10.3 to one, 10.344 to three). Check: 10.34² = 106.9156, close to 107.