Square Root Calculator › Square root of 259
Square Root of 259
√259 ≈ 16.093477
259 is not a perfect square, so √259 is an irrational number that is approximately 16.093477. In simplified radical form, √259 = √259.
√259 lies between √256 = 16 and √289 = 17.
| Number | 259 |
| Square root (decimal) | 16.093477 |
| Simplified radical | √259 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 259
259 = 7 × 37. This is what determines the simplified radical: for √259 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √259 is already in simplest radical form as √259.
Estimating √259 by hand
√259 falls between √256 = 16 and √289 = 17, so it starts at 16.something. The Babylonian method refines that guess by averaging it with 259 ÷ the guess:
| Step | Estimate |
| Start | 16 |
| Step 1 | 16.09375 |
| Step 2 | 16.093477 |
| Step 3 | 16.093477 |
After 3 steps the estimate is 16.093477, matching √259 ≈ 16.093477.
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Frequently asked questions
Is 259 a perfect square?
No, 259 is not a perfect square, so its square root is irrational and approximately 16.093477.
What is the square root of 259 in simplest radical form?
√259 = √259 in simplest radical form.