Square Root Calculator › Square root of 471
Square Root of 471
√471 ≈ 21.702534
471 is not a perfect square, so √471 is an irrational number that is approximately 21.702534. In simplified radical form, √471 = √471.
√471 lies between √441 = 21 and √484 = 22.
| Number | 471 |
| Square root (decimal) | 21.702534 |
| Simplified radical | √471 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 471
471 = 3 × 157. This is what determines the simplified radical: for √471 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √471 is already in simplest radical form as √471.
Estimating √471 by hand
√471 falls between √441 = 21 and √484 = 22, so it starts at 21.something. The Babylonian method refines that guess by averaging it with 471 ÷ the guess:
| Step | Estimate |
| Start | 21 |
| Step 1 | 21.714286 |
| Step 2 | 21.702538 |
| Step 3 | 21.702534 |
After 3 steps the estimate is 21.702534, matching √471 ≈ 21.702534.
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Frequently asked questions
Is 471 a perfect square?
No, 471 is not a perfect square, so its square root is irrational and approximately 21.702534.
What is the square root of 471 in simplest radical form?
√471 = √471 in simplest radical form.