Square Root Calculator › Square root of 397
Square Root of 397
√397 ≈ 19.924859
397 is not a perfect square, so √397 is an irrational number that is approximately 19.924859. In simplified radical form, √397 = √397.
√397 lies between √361 = 19 and √400 = 20.
| Number | 397 |
| Square root (decimal) | 19.924859 |
| Simplified radical | √397 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 397
397 = 397. This is what determines the simplified radical: for √397 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √397 is already in simplest radical form as √397.
Estimating √397 by hand
√397 falls between √361 = 19 and √400 = 20, so it starts at 19.something. The Babylonian method refines that guess by averaging it with 397 ÷ the guess:
| Step | Estimate |
| Start | 19 |
| Step 1 | 19.947368 |
| Step 2 | 19.924872 |
| Step 3 | 19.924859 |
After 3 steps the estimate is 19.924859, matching √397 ≈ 19.924859.
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Frequently asked questions
Is 397 a perfect square?
No, 397 is not a perfect square, so its square root is irrational and approximately 19.924859.
What is the square root of 397 in simplest radical form?
√397 = √397 in simplest radical form.