Square Root Calculator › Square root of 126
Square Root of 126
√126 ≈ 11.224972
126 is not a perfect square, so √126 is an irrational number that is approximately 11.224972. In simplified radical form, √126 = 3√14.
√126 lies between √121 = 11 and √144 = 12.
| Number | 126 |
| Square root (decimal) | 11.224972 |
| Simplified radical | 3√14 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 126
126 = 2 × 3^2 × 7. This is what determines the simplified radical: for √126 you pull out one copy of any prime that appears twice.
Here the paired factor 3 comes out from under the root, leaving √126 = 3√14.
Estimating √126 by hand
√126 falls between √121 = 11 and √144 = 12, so it starts at 11.something. The Babylonian method refines that guess by averaging it with 126 ÷ the guess:
| Step | Estimate |
| Start | 11 |
| Step 1 | 11.227273 |
| Step 2 | 11.224972 |
| Step 3 | 11.224972 |
After 3 steps the estimate is 11.224972, matching √126 ≈ 11.224972.
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Frequently asked questions
Is 126 a perfect square?
No, 126 is not a perfect square, so its square root is irrational and approximately 11.224972.
What is the square root of 126 in simplest radical form?
√126 = 3√14 in simplest radical form.