Square Root Calculator › Square root of 508
Square Root of 508
√508 ≈ 22.538855
508 is not a perfect square, so √508 is an irrational number that is approximately 22.538855. In simplified radical form, √508 = 2√127.
√508 lies between √484 = 22 and √529 = 23.
| Number | 508 |
| Square root (decimal) | 22.538855 |
| Simplified radical | 2√127 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 508
508 = 2^2 × 127. This is what determines the simplified radical: for √508 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √508 = 2√127.
Estimating √508 by hand
√508 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 508 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.545455 |
| Step 2 | 22.538856 |
| Step 3 | 22.538855 |
After 3 steps the estimate is 22.538855, matching √508 ≈ 22.538855.
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Frequently asked questions
Is 508 a perfect square?
No, 508 is not a perfect square, so its square root is irrational and approximately 22.538855.
What is the square root of 508 in simplest radical form?
√508 = 2√127 in simplest radical form.