Square Root Calculator › Square root of 503
Square Root of 503
√503 ≈ 22.427661
503 is not a perfect square, so √503 is an irrational number that is approximately 22.427661. In simplified radical form, √503 = √503.
√503 lies between √484 = 22 and √529 = 23.
| Number | 503 |
| Square root (decimal) | 22.427661 |
| Simplified radical | √503 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 503
503 = 503. This is what determines the simplified radical: for √503 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √503 is already in simplest radical form as √503.
Estimating √503 by hand
√503 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 503 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.431818 |
| Step 2 | 22.427662 |
| Step 3 | 22.427661 |
After 3 steps the estimate is 22.427661, matching √503 ≈ 22.427661.
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Frequently asked questions
Is 503 a perfect square?
No, 503 is not a perfect square, so its square root is irrational and approximately 22.427661.
What is the square root of 503 in simplest radical form?
√503 = √503 in simplest radical form.