Square Root Calculator › Square root of 499
Square Root of 499
√499 ≈ 22.338308
499 is not a perfect square, so √499 is an irrational number that is approximately 22.338308. In simplified radical form, √499 = √499.
√499 lies between √484 = 22 and √529 = 23.
| Number | 499 |
| Square root (decimal) | 22.338308 |
| Simplified radical | √499 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 499
499 = 499. This is what determines the simplified radical: for √499 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √499 is already in simplest radical form as √499.
Estimating √499 by hand
√499 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 499 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.340909 |
| Step 2 | 22.338308 |
| Step 3 | 22.338308 |
After 3 steps the estimate is 22.338308, matching √499 ≈ 22.338308.
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Frequently asked questions
Is 499 a perfect square?
No, 499 is not a perfect square, so its square root is irrational and approximately 22.338308.
What is the square root of 499 in simplest radical form?
√499 = √499 in simplest radical form.