Square Root Calculator › Square root of 447
Square Root of 447
√447 ≈ 21.142375
447 is not a perfect square, so √447 is an irrational number that is approximately 21.142375. In simplified radical form, √447 = √447.
√447 lies between √441 = 21 and √484 = 22.
| Number | 447 |
| Square root (decimal) | 21.142375 |
| Simplified radical | √447 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 447
447 = 3 × 149. This is what determines the simplified radical: for √447 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √447 is already in simplest radical form as √447.
Estimating √447 by hand
√447 falls between √441 = 21 and √484 = 22, so it starts at 21.something. The Babylonian method refines that guess by averaging it with 447 ÷ the guess:
| Step | Estimate |
| Start | 21 |
| Step 1 | 21.142857 |
| Step 2 | 21.142375 |
| Step 3 | 21.142375 |
After 3 steps the estimate is 21.142375, matching √447 ≈ 21.142375.
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Frequently asked questions
Is 447 a perfect square?
No, 447 is not a perfect square, so its square root is irrational and approximately 21.142375.
What is the square root of 447 in simplest radical form?
√447 = √447 in simplest radical form.