Square Root Calculator › Square root of 514
Square Root of 514
√514 ≈ 22.671568
514 is not a perfect square, so √514 is an irrational number that is approximately 22.671568. In simplified radical form, √514 = √514.
√514 lies between √484 = 22 and √529 = 23.
| Number | 514 |
| Square root (decimal) | 22.671568 |
| Simplified radical | √514 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 514
514 = 2 × 257. This is what determines the simplified radical: for √514 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √514 is already in simplest radical form as √514.
Estimating √514 by hand
√514 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 514 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.681818 |
| Step 2 | 22.67157 |
| Step 3 | 22.671568 |
After 3 steps the estimate is 22.671568, matching √514 ≈ 22.671568.
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Frequently asked questions
Is 514 a perfect square?
No, 514 is not a perfect square, so its square root is irrational and approximately 22.671568.
What is the square root of 514 in simplest radical form?
√514 = √514 in simplest radical form.