Square Root Calculator › Square root of 477
Square Root of 477
√477 ≈ 21.84033
477 is not a perfect square, so √477 is an irrational number that is approximately 21.84033. In simplified radical form, √477 = 3√53.
√477 lies between √441 = 21 and √484 = 22.
| Number | 477 |
| Square root (decimal) | 21.84033 |
| Simplified radical | 3√53 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 477
477 = 3^2 × 53. This is what determines the simplified radical: for √477 you pull out one copy of any prime that appears twice.
Here the paired factor 3 comes out from under the root, leaving √477 = 3√53.
Estimating √477 by hand
√477 falls between √441 = 21 and √484 = 22, so it starts at 21.something. The Babylonian method refines that guess by averaging it with 477 ÷ the guess:
| Step | Estimate |
| Start | 21 |
| Step 1 | 21.857143 |
| Step 2 | 21.840336 |
| Step 3 | 21.84033 |
After 3 steps the estimate is 21.84033, matching √477 ≈ 21.84033.
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Frequently asked questions
Is 477 a perfect square?
No, 477 is not a perfect square, so its square root is irrational and approximately 21.84033.
What is the square root of 477 in simplest radical form?
√477 = 3√53 in simplest radical form.