Square Root Calculator › Square root of 381
Square Root of 381
√381 ≈ 19.519221
381 is not a perfect square, so √381 is an irrational number that is approximately 19.519221. In simplified radical form, √381 = √381.
√381 lies between √361 = 19 and √400 = 20.
| Number | 381 |
| Square root (decimal) | 19.519221 |
| Simplified radical | √381 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 381
381 = 3 × 127. This is what determines the simplified radical: for √381 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √381 is already in simplest radical form as √381.
Estimating √381 by hand
√381 falls between √361 = 19 and √400 = 20, so it starts at 19.something. The Babylonian method refines that guess by averaging it with 381 ÷ the guess:
| Step | Estimate |
| Start | 19 |
| Step 1 | 19.526316 |
| Step 2 | 19.519223 |
| Step 3 | 19.519221 |
After 3 steps the estimate is 19.519221, matching √381 ≈ 19.519221.
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Frequently asked questions
Is 381 a perfect square?
No, 381 is not a perfect square, so its square root is irrational and approximately 19.519221.
What is the square root of 381 in simplest radical form?
√381 = √381 in simplest radical form.