Square Root Calculator › Square root of 378
Square Root of 378
√378 ≈ 19.442222
378 is not a perfect square, so √378 is an irrational number that is approximately 19.442222. In simplified radical form, √378 = 3√42.
√378 lies between √361 = 19 and √400 = 20.
| Number | 378 |
| Square root (decimal) | 19.442222 |
| Simplified radical | 3√42 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 378
378 = 2 × 3^3 × 7. This is what determines the simplified radical: for √378 you pull out one copy of any prime that appears twice.
Here the paired factor 3 comes out from under the root, leaving √378 = 3√42.
Estimating √378 by hand
√378 falls between √361 = 19 and √400 = 20, so it starts at 19.something. The Babylonian method refines that guess by averaging it with 378 ÷ the guess:
| Step | Estimate |
| Start | 19 |
| Step 1 | 19.447368 |
| Step 2 | 19.442223 |
| Step 3 | 19.442222 |
After 3 steps the estimate is 19.442222, matching √378 ≈ 19.442222.
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Frequently asked questions
Is 378 a perfect square?
No, 378 is not a perfect square, so its square root is irrational and approximately 19.442222.
What is the square root of 378 in simplest radical form?
√378 = 3√42 in simplest radical form.