Square Root Calculator › Square root of 459
Square Root of 459
√459 ≈ 21.424285
459 is not a perfect square, so √459 is an irrational number that is approximately 21.424285. In simplified radical form, √459 = 3√51.
√459 lies between √441 = 21 and √484 = 22.
| Number | 459 |
| Square root (decimal) | 21.424285 |
| Simplified radical | 3√51 |
| Perfect square? | No |
| Rational? | No |
Calculate another square root
Prime factorization of 459
459 = 3^3 × 17. This is what determines the simplified radical: for √459 you pull out one copy of any prime that appears twice.
Here the paired factor 3 comes out from under the root, leaving √459 = 3√51.
Estimating √459 by hand
√459 falls between √441 = 21 and √484 = 22, so it starts at 21.something. The Babylonian method refines that guess by averaging it with 459 ÷ the guess:
| Step | Estimate |
| Start | 21 |
| Step 1 | 21.428571 |
| Step 2 | 21.424286 |
| Step 3 | 21.424285 |
After 3 steps the estimate is 21.424285, matching √459 ≈ 21.424285.
Nearby square roots
Frequently asked questions
Is 459 a perfect square?
No, 459 is not a perfect square, so its square root is irrational and approximately 21.424285.
What is the square root of 459 in simplest radical form?
√459 = 3√51 in simplest radical form.