Square Root Calculator › Square root of 107
Square Root of 107
√107 ≈ 10.34408
107 is not a perfect square, so √107 is an irrational number that is approximately 10.34408. In simplified radical form, √107 = √107.
√107 lies between √100 = 10 and √121 = 11.
| Number | 107 |
| Square root (decimal) | 10.34408 |
| Simplified radical | √107 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 107
107 = 107. This is what determines the simplified radical: for √107 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √107 is already in simplest radical form as √107.
Estimating √107 by hand
√107 falls between √100 = 10 and √121 = 11, so it starts at 10.something. The Babylonian method refines that guess by averaging it with 107 ÷ the guess:
| Step | Estimate |
| Start | 10 |
| Step 1 | 10.35 |
| Step 2 | 10.344082 |
| Step 3 | 10.34408 |
After 3 steps the estimate is 10.34408, matching √107 ≈ 10.34408.
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Frequently asked questions
Is 107 a perfect square?
No, 107 is not a perfect square, so its square root is irrational and approximately 10.34408.
What is the square root of 107 in simplest radical form?
√107 = √107 in simplest radical form.