Square Root Calculator › Square root of 319
Square Root of 319
√319 ≈ 17.860571
319 is not a perfect square, so √319 is an irrational number that is approximately 17.860571. In simplified radical form, √319 = √319.
√319 lies between √289 = 17 and √324 = 18.
| Number | 319 |
| Square root (decimal) | 17.860571 |
| Simplified radical | √319 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 319
319 = 11 × 29. This is what determines the simplified radical: for √319 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √319 is already in simplest radical form as √319.
Estimating √319 by hand
√319 falls between √289 = 17 and √324 = 18, so it starts at 17.something. The Babylonian method refines that guess by averaging it with 319 ÷ the guess:
| Step | Estimate |
| Start | 17 |
| Step 1 | 17.882353 |
| Step 2 | 17.860584 |
| Step 3 | 17.860571 |
After 3 steps the estimate is 17.860571, matching √319 ≈ 17.860571.
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Frequently asked questions
Is 319 a perfect square?
No, 319 is not a perfect square, so its square root is irrational and approximately 17.860571.
What is the square root of 319 in simplest radical form?
√319 = √319 in simplest radical form.