Square Root Calculator › Square root of 138
Square Root of 138
√138 ≈ 11.74734
138 is not a perfect square, so √138 is an irrational number that is approximately 11.74734. In simplified radical form, √138 = √138.
√138 lies between √121 = 11 and √144 = 12.
| Number | 138 |
| Square root (decimal) | 11.74734 |
| Simplified radical | √138 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 138
138 = 2 × 3 × 23. This is what determines the simplified radical: for √138 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √138 is already in simplest radical form as √138.
Estimating √138 by hand
√138 falls between √121 = 11 and √144 = 12, so it starts at 11.something. The Babylonian method refines that guess by averaging it with 138 ÷ the guess:
| Step | Estimate |
| Start | 11 |
| Step 1 | 11.772727 |
| Step 2 | 11.747367 |
| Step 3 | 11.74734 |
After 3 steps the estimate is 11.74734, matching √138 ≈ 11.74734.
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Frequently asked questions
Is 138 a perfect square?
No, 138 is not a perfect square, so its square root is irrational and approximately 11.74734.
What is the square root of 138 in simplest radical form?
√138 = √138 in simplest radical form.