Square Root Calculator › Square root of 257
Square Root of 257
√257 ≈ 16.03122
257 is not a perfect square, so √257 is an irrational number that is approximately 16.03122. In simplified radical form, √257 = √257.
√257 lies between √256 = 16 and √289 = 17.
| Number | 257 |
| Square root (decimal) | 16.03122 |
| Simplified radical | √257 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 257
257 = 257. This is what determines the simplified radical: for √257 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √257 is already in simplest radical form as √257.
Estimating √257 by hand
√257 falls between √256 = 16 and √289 = 17, so it starts at 16.something. The Babylonian method refines that guess by averaging it with 257 ÷ the guess:
| Step | Estimate |
| Start | 16 |
| Step 1 | 16.03125 |
| Step 2 | 16.03122 |
| Step 3 | 16.03122 |
After 3 steps the estimate is 16.03122, matching √257 ≈ 16.03122.
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Frequently asked questions
Is 257 a perfect square?
No, 257 is not a perfect square, so its square root is irrational and approximately 16.03122.
What is the square root of 257 in simplest radical form?
√257 = √257 in simplest radical form.