Square Root Calculator › Square root of 464
Square Root of 464
√464 ≈ 21.540659
464 is not a perfect square, so √464 is an irrational number that is approximately 21.540659. In simplified radical form, √464 = 4√29.
√464 lies between √441 = 21 and √484 = 22.
| Number | 464 |
| Square root (decimal) | 21.540659 |
| Simplified radical | 4√29 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 464
464 = 2^4 × 29. This is what determines the simplified radical: for √464 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √464 = 4√29.
Estimating √464 by hand
√464 falls between √441 = 21 and √484 = 22, so it starts at 21.something. The Babylonian method refines that guess by averaging it with 464 ÷ the guess:
| Step | Estimate |
| Start | 21 |
| Step 1 | 21.547619 |
| Step 2 | 21.54066 |
| Step 3 | 21.540659 |
After 3 steps the estimate is 21.540659, matching √464 ≈ 21.540659.
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Frequently asked questions
Is 464 a perfect square?
No, 464 is not a perfect square, so its square root is irrational and approximately 21.540659.
What is the square root of 464 in simplest radical form?
√464 = 4√29 in simplest radical form.