Square Root Calculator › Square root of 307
Square Root of 307
√307 ≈ 17.521415
307 is not a perfect square, so √307 is an irrational number that is approximately 17.521415. In simplified radical form, √307 = √307.
√307 lies between √289 = 17 and √324 = 18.
| Number | 307 |
| Square root (decimal) | 17.521415 |
| Simplified radical | √307 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 307
307 = 307. This is what determines the simplified radical: for √307 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √307 is already in simplest radical form as √307.
Estimating √307 by hand
√307 falls between √289 = 17 and √324 = 18, so it starts at 17.something. The Babylonian method refines that guess by averaging it with 307 ÷ the guess:
| Step | Estimate |
| Start | 17 |
| Step 1 | 17.529412 |
| Step 2 | 17.521417 |
| Step 3 | 17.521415 |
After 3 steps the estimate is 17.521415, matching √307 ≈ 17.521415.
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Frequently asked questions
Is 307 a perfect square?
No, 307 is not a perfect square, so its square root is irrational and approximately 17.521415.
What is the square root of 307 in simplest radical form?
√307 = √307 in simplest radical form.