Square Root Calculator › Square root of 507
Square Root of 507
√507 ≈ 22.51666
507 is not a perfect square, so √507 is an irrational number that is approximately 22.51666. In simplified radical form, √507 = 13√3.
√507 lies between √484 = 22 and √529 = 23.
| Number | 507 |
| Square root (decimal) | 22.51666 |
| Simplified radical | 13√3 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 507
507 = 3 × 13^2. This is what determines the simplified radical: for √507 you pull out one copy of any prime that appears twice.
Here the paired factor 13 comes out from under the root, leaving √507 = 13√3.
Estimating √507 by hand
√507 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 507 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.522727 |
| Step 2 | 22.516661 |
| Step 3 | 22.51666 |
After 3 steps the estimate is 22.51666, matching √507 ≈ 22.51666.
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Frequently asked questions
Is 507 a perfect square?
No, 507 is not a perfect square, so its square root is irrational and approximately 22.51666.
What is the square root of 507 in simplest radical form?
√507 = 13√3 in simplest radical form.