Square Root Calculator › Square root of 523
Square Root of 523
√523 ≈ 22.869193
523 is not a perfect square, so √523 is an irrational number that is approximately 22.869193. In simplified radical form, √523 = √523.
√523 lies between √484 = 22 and √529 = 23.
| Number | 523 |
| Square root (decimal) | 22.869193 |
| Simplified radical | √523 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 523
523 = 523. This is what determines the simplified radical: for √523 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √523 is already in simplest radical form as √523.
Estimating √523 by hand
√523 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 523 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.886364 |
| Step 2 | 22.8692 |
| Step 3 | 22.869193 |
After 3 steps the estimate is 22.869193, matching √523 ≈ 22.869193.
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Frequently asked questions
Is 523 a perfect square?
No, 523 is not a perfect square, so its square root is irrational and approximately 22.869193.
What is the square root of 523 in simplest radical form?
√523 = √523 in simplest radical form.