Square Root Calculator › Square root of 117
Square Root of 117
√117 ≈ 10.816654
117 is not a perfect square, so √117 is an irrational number that is approximately 10.816654. In simplified radical form, √117 = 3√13.
√117 lies between √100 = 10 and √121 = 11.
| Number | 117 |
| Square root (decimal) | 10.816654 |
| Simplified radical | 3√13 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 117
117 = 3^2 × 13. This is what determines the simplified radical: for √117 you pull out one copy of any prime that appears twice.
Here the paired factor 3 comes out from under the root, leaving √117 = 3√13.
Estimating √117 by hand
√117 falls between √100 = 10 and √121 = 11, so it starts at 10.something. The Babylonian method refines that guess by averaging it with 117 ÷ the guess:
| Step | Estimate |
| Start | 10 |
| Step 1 | 10.85 |
| Step 2 | 10.816705 |
| Step 3 | 10.816654 |
After 3 steps the estimate is 10.816654, matching √117 ≈ 10.816654.
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Frequently asked questions
Is 117 a perfect square?
No, 117 is not a perfect square, so its square root is irrational and approximately 10.816654.
What is the square root of 117 in simplest radical form?
√117 = 3√13 in simplest radical form.