Square Root Calculator › Square root of 493
Square Root of 493
√493 ≈ 22.203603
493 is not a perfect square, so √493 is an irrational number that is approximately 22.203603. In simplified radical form, √493 = √493.
√493 lies between √484 = 22 and √529 = 23.
| Number | 493 |
| Square root (decimal) | 22.203603 |
| Simplified radical | √493 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 493
493 = 17 × 29. This is what determines the simplified radical: for √493 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √493 is already in simplest radical form as √493.
Estimating √493 by hand
√493 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 493 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.204545 |
| Step 2 | 22.203603 |
| Step 3 | 22.203603 |
After 3 steps the estimate is 22.203603, matching √493 ≈ 22.203603.
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Frequently asked questions
Is 493 a perfect square?
No, 493 is not a perfect square, so its square root is irrational and approximately 22.203603.
What is the square root of 493 in simplest radical form?
√493 = √493 in simplest radical form.