Square Root Calculator › Square root of 111
Square Root of 111
√111 ≈ 10.535654
111 is not a perfect square, so √111 is an irrational number that is approximately 10.535654. In simplified radical form, √111 = √111.
√111 lies between √100 = 10 and √121 = 11.
| Number | 111 |
| Square root (decimal) | 10.535654 |
| Simplified radical | √111 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 111
111 = 3 × 37. This is what determines the simplified radical: for √111 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √111 is already in simplest radical form as √111.
Estimating √111 by hand
√111 falls between √100 = 10 and √121 = 11, so it starts at 10.something. The Babylonian method refines that guess by averaging it with 111 ÷ the guess:
| Step | Estimate |
| Start | 10 |
| Step 1 | 10.55 |
| Step 2 | 10.535664 |
| Step 3 | 10.535654 |
After 3 steps the estimate is 10.535654, matching √111 ≈ 10.535654.
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Frequently asked questions
Is 111 a perfect square?
No, 111 is not a perfect square, so its square root is irrational and approximately 10.535654.
What is the square root of 111 in simplest radical form?
√111 = √111 in simplest radical form.