Square Root Calculator › Square root of 500
Square Root of 500
√500 ≈ 22.36068
500 is not a perfect square, so √500 is an irrational number that is approximately 22.36068. In simplified radical form, √500 = 10√5.
√500 lies between √484 = 22 and √529 = 23.
| Number | 500 |
| Square root (decimal) | 22.36068 |
| Simplified radical | 10√5 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 500
500 = 2^2 × 5^3. This is what determines the simplified radical: for √500 you pull out one copy of any prime that appears twice.
Here the paired factors 2 and 5 come out from under the root, leaving √500 = 10√5.
Estimating √500 by hand
√500 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 500 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.363636 |
| Step 2 | 22.36068 |
| Step 3 | 22.36068 |
After 3 steps the estimate is 22.36068, matching √500 ≈ 22.36068.
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Frequently asked questions
Is 500 a perfect square?
No, 500 is not a perfect square, so its square root is irrational and approximately 22.36068.
What is the square root of 500 in simplest radical form?
√500 = 10√5 in simplest radical form.