Square Root Calculator › Square root of 478
Square Root of 478
√478 ≈ 21.863211
478 is not a perfect square, so √478 is an irrational number that is approximately 21.863211. In simplified radical form, √478 = √478.
√478 lies between √441 = 21 and √484 = 22.
| Number | 478 |
| Square root (decimal) | 21.863211 |
| Simplified radical | √478 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 478
478 = 2 × 239. This is what determines the simplified radical: for √478 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √478 is already in simplest radical form as √478.
Estimating √478 by hand
√478 falls between √441 = 21 and √484 = 22, so it starts at 21.something. The Babylonian method refines that guess by averaging it with 478 ÷ the guess:
| Step | Estimate |
| Start | 21 |
| Step 1 | 21.880952 |
| Step 2 | 21.863218 |
| Step 3 | 21.863211 |
After 3 steps the estimate is 21.863211, matching √478 ≈ 21.863211.
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Frequently asked questions
Is 478 a perfect square?
No, 478 is not a perfect square, so its square root is irrational and approximately 21.863211.
What is the square root of 478 in simplest radical form?
√478 = √478 in simplest radical form.