Square Root Calculator › Square root of 137
Square Root of 137
√137 ≈ 11.7047
137 is not a perfect square, so √137 is an irrational number that is approximately 11.7047. In simplified radical form, √137 = √137.
√137 lies between √121 = 11 and √144 = 12.
| Number | 137 |
| Square root (decimal) | 11.7047 |
| Simplified radical | √137 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 137
137 = 137. This is what determines the simplified radical: for √137 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √137 is already in simplest radical form as √137.
Estimating √137 by hand
√137 falls between √121 = 11 and √144 = 12, so it starts at 11.something. The Babylonian method refines that guess by averaging it with 137 ÷ the guess:
| Step | Estimate |
| Start | 11 |
| Step 1 | 11.727273 |
| Step 2 | 11.704722 |
| Step 3 | 11.7047 |
After 3 steps the estimate is 11.7047, matching √137 ≈ 11.7047.
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Frequently asked questions
Is 137 a perfect square?
No, 137 is not a perfect square, so its square root is irrational and approximately 11.7047.
What is the square root of 137 in simplest radical form?
√137 = √137 in simplest radical form.