Square Root Calculator › Square root of 585
Square Root of 585
√585 ≈ 24.186773
585 is not a perfect square, so √585 is an irrational number that is approximately 24.186773. In simplified radical form, √585 = 3√65.
√585 lies between √576 = 24 and √625 = 25.
| Number | 585 |
| Square root (decimal) | 24.186773 |
| Simplified radical | 3√65 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 585
585 = 3^2 × 5 × 13. This is what determines the simplified radical: for √585 you pull out one copy of any prime that appears twice.
Here the paired factor 3 comes out from under the root, leaving √585 = 3√65.
Estimating √585 by hand
√585 falls between √576 = 24 and √625 = 25, so it starts at 24.something. The Babylonian method refines that guess by averaging it with 585 ÷ the guess:
| Step | Estimate |
| Start | 24 |
| Step 1 | 24.1875 |
| Step 2 | 24.186773 |
| Step 3 | 24.186773 |
After 3 steps the estimate is 24.186773, matching √585 ≈ 24.186773.
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Frequently asked questions
Is 585 a perfect square?
No, 585 is not a perfect square, so its square root is irrational and approximately 24.186773.
What is the square root of 585 in simplest radical form?
√585 = 3√65 in simplest radical form.