Square Root Calculator › Square root of 513
Square Root of 513
√513 ≈ 22.649503
513 is not a perfect square, so √513 is an irrational number that is approximately 22.649503. In simplified radical form, √513 = 3√57.
√513 lies between √484 = 22 and √529 = 23.
| Number | 513 |
| Square root (decimal) | 22.649503 |
| Simplified radical | 3√57 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 513
513 = 3^3 × 19. This is what determines the simplified radical: for √513 you pull out one copy of any prime that appears twice.
Here the paired factor 3 comes out from under the root, leaving √513 = 3√57.
Estimating √513 by hand
√513 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 513 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.659091 |
| Step 2 | 22.649505 |
| Step 3 | 22.649503 |
After 3 steps the estimate is 22.649503, matching √513 ≈ 22.649503.
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Frequently asked questions
Is 513 a perfect square?
No, 513 is not a perfect square, so its square root is irrational and approximately 22.649503.
What is the square root of 513 in simplest radical form?
√513 = 3√57 in simplest radical form.