Square Root Calculator › Square root of 384
Square Root of 384
√384 ≈ 19.595918
384 is not a perfect square, so √384 is an irrational number that is approximately 19.595918. In simplified radical form, √384 = 8√6.
√384 lies between √361 = 19 and √400 = 20.
| Number | 384 |
| Square root (decimal) | 19.595918 |
| Simplified radical | 8√6 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 384
384 = 2^7 × 3. This is what determines the simplified radical: for √384 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √384 = 8√6.
Estimating √384 by hand
√384 falls between √361 = 19 and √400 = 20, so it starts at 19.something. The Babylonian method refines that guess by averaging it with 384 ÷ the guess:
| Step | Estimate |
| Start | 19 |
| Step 1 | 19.605263 |
| Step 2 | 19.59592 |
| Step 3 | 19.595918 |
After 3 steps the estimate is 19.595918, matching √384 ≈ 19.595918.
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Frequently asked questions
Is 384 a perfect square?
No, 384 is not a perfect square, so its square root is irrational and approximately 19.595918.
What is the square root of 384 in simplest radical form?
√384 = 8√6 in simplest radical form.