Square Root Calculator › Square root of 522
Square Root of 522
√522 ≈ 22.847319
522 is not a perfect square, so √522 is an irrational number that is approximately 22.847319. In simplified radical form, √522 = 3√58.
√522 lies between √484 = 22 and √529 = 23.
| Number | 522 |
| Square root (decimal) | 22.847319 |
| Simplified radical | 3√58 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 522
522 = 2 × 3^2 × 29. This is what determines the simplified radical: for √522 you pull out one copy of any prime that appears twice.
Here the paired factor 3 comes out from under the root, leaving √522 = 3√58.
Estimating √522 by hand
√522 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 522 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.863636 |
| Step 2 | 22.847325 |
| Step 3 | 22.847319 |
After 3 steps the estimate is 22.847319, matching √522 ≈ 22.847319.
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Frequently asked questions
Is 522 a perfect square?
No, 522 is not a perfect square, so its square root is irrational and approximately 22.847319.
What is the square root of 522 in simplest radical form?
√522 = 3√58 in simplest radical form.