Square Root Calculator › Square root of 123
Square Root of 123
√123 ≈ 11.090537
123 is not a perfect square, so √123 is an irrational number that is approximately 11.090537. In simplified radical form, √123 = √123.
√123 lies between √121 = 11 and √144 = 12.
| Number | 123 |
| Square root (decimal) | 11.090537 |
| Simplified radical | √123 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 123
123 = 3 × 41. This is what determines the simplified radical: for √123 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √123 is already in simplest radical form as √123.
Estimating √123 by hand
√123 falls between √121 = 11 and √144 = 12, so it starts at 11.something. The Babylonian method refines that guess by averaging it with 123 ÷ the guess:
| Step | Estimate |
| Start | 11 |
| Step 1 | 11.090909 |
| Step 2 | 11.090537 |
| Step 3 | 11.090537 |
After 3 steps the estimate is 11.090537, matching √123 ≈ 11.090537.
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Frequently asked questions
Is 123 a perfect square?
No, 123 is not a perfect square, so its square root is irrational and approximately 11.090537.
What is the square root of 123 in simplest radical form?
√123 = √123 in simplest radical form.