Square Root Calculator › Square root of 105
Square Root of 105
√105 ≈ 10.246951
105 is not a perfect square, so √105 is an irrational number that is approximately 10.246951. In simplified radical form, √105 = √105.
√105 lies between √100 = 10 and √121 = 11.
| Number | 105 |
| Square root (decimal) | 10.246951 |
| Simplified radical | √105 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 105
105 = 3 × 5 × 7. This is what determines the simplified radical: for √105 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √105 is already in simplest radical form as √105.
Estimating √105 by hand
√105 falls between √100 = 10 and √121 = 11, so it starts at 10.something. The Babylonian method refines that guess by averaging it with 105 ÷ the guess:
| Step | Estimate |
| Start | 10 |
| Step 1 | 10.25 |
| Step 2 | 10.246951 |
| Step 3 | 10.246951 |
After 3 steps the estimate is 10.246951, matching √105 ≈ 10.246951.
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Frequently asked questions
Is 105 a perfect square?
No, 105 is not a perfect square, so its square root is irrational and approximately 10.246951.
What is the square root of 105 in simplest radical form?
√105 = √105 in simplest radical form.