Square Root Calculator › Square root of 538
Square Root of 538
√538 ≈ 23.194827
538 is not a perfect square, so √538 is an irrational number that is approximately 23.194827. In simplified radical form, √538 = √538.
√538 lies between √529 = 23 and √576 = 24.
| Number | 538 |
| Square root (decimal) | 23.194827 |
| Simplified radical | √538 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 538
538 = 2 × 269. This is what determines the simplified radical: for √538 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √538 is already in simplest radical form as √538.
Estimating √538 by hand
√538 falls between √529 = 23 and √576 = 24, so it starts at 23.something. The Babylonian method refines that guess by averaging it with 538 ÷ the guess:
| Step | Estimate |
| Start | 23 |
| Step 1 | 23.195652 |
| Step 2 | 23.194827 |
| Step 3 | 23.194827 |
After 3 steps the estimate is 23.194827, matching √538 ≈ 23.194827.
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Frequently asked questions
Is 538 a perfect square?
No, 538 is not a perfect square, so its square root is irrational and approximately 23.194827.
What is the square root of 538 in simplest radical form?
√538 = √538 in simplest radical form.