Square Root Calculator › Square root of 106
Square Root of 106
√106 ≈ 10.29563
106 is not a perfect square, so √106 is an irrational number that is approximately 10.29563. In simplified radical form, √106 = √106.
√106 lies between √100 = 10 and √121 = 11.
| Number | 106 |
| Square root (decimal) | 10.29563 |
| Simplified radical | √106 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 106
106 = 2 × 53. This is what determines the simplified radical: for √106 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √106 is already in simplest radical form as √106.
Estimating √106 by hand
√106 falls between √100 = 10 and √121 = 11, so it starts at 10.something. The Babylonian method refines that guess by averaging it with 106 ÷ the guess:
| Step | Estimate |
| Start | 10 |
| Step 1 | 10.3 |
| Step 2 | 10.295631 |
| Step 3 | 10.29563 |
After 3 steps the estimate is 10.29563, matching √106 ≈ 10.29563.
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Frequently asked questions
Is 106 a perfect square?
No, 106 is not a perfect square, so its square root is irrational and approximately 10.29563.
What is the square root of 106 in simplest radical form?
√106 = √106 in simplest radical form.