Square Root Calculator › Square root of 463
Square Root of 463
√463 ≈ 21.517435
463 is not a perfect square, so √463 is an irrational number that is approximately 21.517435. In simplified radical form, √463 = √463.
√463 lies between √441 = 21 and √484 = 22.
| Number | 463 |
| Square root (decimal) | 21.517435 |
| Simplified radical | √463 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 463
463 = 463. This is what determines the simplified radical: for √463 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √463 is already in simplest radical form as √463.
Estimating √463 by hand
√463 falls between √441 = 21 and √484 = 22, so it starts at 21.something. The Babylonian method refines that guess by averaging it with 463 ÷ the guess:
| Step | Estimate |
| Start | 21 |
| Step 1 | 21.52381 |
| Step 2 | 21.517436 |
| Step 3 | 21.517435 |
After 3 steps the estimate is 21.517435, matching √463 ≈ 21.517435.
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Frequently asked questions
Is 463 a perfect square?
No, 463 is not a perfect square, so its square root is irrational and approximately 21.517435.
What is the square root of 463 in simplest radical form?
√463 = √463 in simplest radical form.