Square Root Calculator › Square root of 393
Square Root of 393
√393 ≈ 19.824228
393 is not a perfect square, so √393 is an irrational number that is approximately 19.824228. In simplified radical form, √393 = √393.
√393 lies between √361 = 19 and √400 = 20.
| Number | 393 |
| Square root (decimal) | 19.824228 |
| Simplified radical | √393 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 393
393 = 3 × 131. This is what determines the simplified radical: for √393 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √393 is already in simplest radical form as √393.
Estimating √393 by hand
√393 falls between √361 = 19 and √400 = 20, so it starts at 19.something. The Babylonian method refines that guess by averaging it with 393 ÷ the guess:
| Step | Estimate |
| Start | 19 |
| Step 1 | 19.842105 |
| Step 2 | 19.824236 |
| Step 3 | 19.824228 |
After 3 steps the estimate is 19.824228, matching √393 ≈ 19.824228.
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Frequently asked questions
Is 393 a perfect square?
No, 393 is not a perfect square, so its square root is irrational and approximately 19.824228.
What is the square root of 393 in simplest radical form?
√393 = √393 in simplest radical form.