Square Root Calculator › Square root of 506
Square Root of 506
√506 ≈ 22.494444
506 is not a perfect square, so √506 is an irrational number that is approximately 22.494444. In simplified radical form, √506 = √506.
√506 lies between √484 = 22 and √529 = 23.
| Number | 506 |
| Square root (decimal) | 22.494444 |
| Simplified radical | √506 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 506
506 = 2 × 11 × 23. This is what determines the simplified radical: for √506 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √506 is already in simplest radical form as √506.
Estimating √506 by hand
√506 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 506 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.5 |
| Step 2 | 22.494444 |
| Step 3 | 22.494444 |
After 3 steps the estimate is 22.494444, matching √506 ≈ 22.494444.
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Frequently asked questions
Is 506 a perfect square?
No, 506 is not a perfect square, so its square root is irrational and approximately 22.494444.
What is the square root of 506 in simplest radical form?
√506 = √506 in simplest radical form.