Square Root Calculator › Square root of 501
Square Root of 501
√501 ≈ 22.383029
501 is not a perfect square, so √501 is an irrational number that is approximately 22.383029. In simplified radical form, √501 = √501.
√501 lies between √484 = 22 and √529 = 23.
| Number | 501 |
| Square root (decimal) | 22.383029 |
| Simplified radical | √501 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 501
501 = 3 × 167. This is what determines the simplified radical: for √501 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √501 is already in simplest radical form as √501.
Estimating √501 by hand
√501 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 501 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.386364 |
| Step 2 | 22.38303 |
| Step 3 | 22.383029 |
After 3 steps the estimate is 22.383029, matching √501 ≈ 22.383029.
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Frequently asked questions
Is 501 a perfect square?
No, 501 is not a perfect square, so its square root is irrational and approximately 22.383029.
What is the square root of 501 in simplest radical form?
√501 = √501 in simplest radical form.