Square Root Calculator › Square root of 512
Square Root of 512
√512 ≈ 22.627417
512 is not a perfect square, so √512 is an irrational number that is approximately 22.627417. In simplified radical form, √512 = 16√2.
√512 lies between √484 = 22 and √529 = 23.
| Number | 512 |
| Square root (decimal) | 22.627417 |
| Simplified radical | 16√2 |
| Perfect square? | No |
| Rational? | No |
Calculate another square root
Prime factorization of 512
512 = 2^9. This is what determines the simplified radical: for √512 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √512 = 16√2.
Estimating √512 by hand
√512 falls between √484 = 22 and √529 = 23, so it starts at 22.something. The Babylonian method refines that guess by averaging it with 512 ÷ the guess:
| Step | Estimate |
| Start | 22 |
| Step 1 | 22.636364 |
| Step 2 | 22.627419 |
| Step 3 | 22.627417 |
After 3 steps the estimate is 22.627417, matching √512 ≈ 22.627417.
Nearby square roots
Frequently asked questions
Is 512 a perfect square?
No, 512 is not a perfect square, so its square root is irrational and approximately 22.627417.
What is the square root of 512 in simplest radical form?
√512 = 16√2 in simplest radical form.