Square Root Calculator › Square root of 109
Square Root of 109
√109 ≈ 10.440307
109 is not a perfect square, so √109 is an irrational number that is approximately 10.440307. In simplified radical form, √109 = √109.
√109 lies between √100 = 10 and √121 = 11.
| Number | 109 |
| Square root (decimal) | 10.440307 |
| Simplified radical | √109 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 109
109 = 109. This is what determines the simplified radical: for √109 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √109 is already in simplest radical form as √109.
Estimating √109 by hand
√109 falls between √100 = 10 and √121 = 11, so it starts at 10.something. The Babylonian method refines that guess by averaging it with 109 ÷ the guess:
| Step | Estimate |
| Start | 10 |
| Step 1 | 10.45 |
| Step 2 | 10.440311 |
| Step 3 | 10.440307 |
After 3 steps the estimate is 10.440307, matching √109 ≈ 10.440307.
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Frequently asked questions
Is 109 a perfect square?
No, 109 is not a perfect square, so its square root is irrational and approximately 10.440307.
What is the square root of 109 in simplest radical form?
√109 = √109 in simplest radical form.