Square Root Calculator › Square root of 521
Square Root of 521
√521 ≈ 22.825424
521 is not a perfect square, so √521 is an irrational number that is approximately 22.825424. In simplified radical form, √521 = √521.
√521 lies between √484 = 22 and √529 = 23.
| Number | 521 |
| Square root (decimal) | 22.825424 |
| Simplified radical | √521 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 521
521 = 521. This is what determines the simplified radical: for √521 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √521 is already in simplest radical form as √521.
Estimating √521 by hand
√521 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 521 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.840909 |
| Step 2 | 22.82543 |
| Step 3 | 22.825424 |
After 3 steps the estimate is 22.825424, matching √521 ≈ 22.825424.
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Frequently asked questions
Is 521 a perfect square?
No, 521 is not a perfect square, so its square root is irrational and approximately 22.825424.
What is the square root of 521 in simplest radical form?
√521 = √521 in simplest radical form.