Square Root Calculator › Square root of 527
Square Root of 527
√527 ≈ 22.956481
527 is not a perfect square, so √527 is an irrational number that is approximately 22.956481. In simplified radical form, √527 = √527.
√527 lies between √484 = 22 and √529 = 23.
| Number | 527 |
| Square root (decimal) | 22.956481 |
| Simplified radical | √527 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 527
527 = 17 × 31. This is what determines the simplified radical: for √527 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √527 is already in simplest radical form as √527.
Estimating √527 by hand
√527 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 527 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.977273 |
| Step 2 | 22.95649 |
| Step 3 | 22.956481 |
After 3 steps the estimate is 22.956481, matching √527 ≈ 22.956481.
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Frequently asked questions
Is 527 a perfect square?
No, 527 is not a perfect square, so its square root is irrational and approximately 22.956481.
What is the square root of 527 in simplest radical form?
√527 = √527 in simplest radical form.