Square Root Calculator › Square root of 313
Square Root of 313
√313 ≈ 17.691806
313 is not a perfect square, so √313 is an irrational number that is approximately 17.691806. In simplified radical form, √313 = √313.
√313 lies between √289 = 17 and √324 = 18.
| Number | 313 |
| Square root (decimal) | 17.691806 |
| Simplified radical | √313 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 313
313 = 313. This is what determines the simplified radical: for √313 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √313 is already in simplest radical form as √313.
Estimating √313 by hand
√313 falls between √289 = 17 and √324 = 18, so it starts at 17.something. The Babylonian method refines that guess by averaging it with 313 ÷ the guess:
| Step | Estimate |
| Start | 17 |
| Step 1 | 17.705882 |
| Step 2 | 17.691812 |
| Step 3 | 17.691806 |
After 3 steps the estimate is 17.691806, matching √313 ≈ 17.691806.
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Frequently asked questions
Is 313 a perfect square?
No, 313 is not a perfect square, so its square root is irrational and approximately 17.691806.
What is the square root of 313 in simplest radical form?
√313 = √313 in simplest radical form.