Square Root Calculator › Square root of 466
Square Root of 466
√466 ≈ 21.587033
466 is not a perfect square, so √466 is an irrational number that is approximately 21.587033. In simplified radical form, √466 = √466.
√466 lies between √441 = 21 and √484 = 22.
| Number | 466 |
| Square root (decimal) | 21.587033 |
| Simplified radical | √466 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 466
466 = 2 × 233. This is what determines the simplified radical: for √466 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √466 is already in simplest radical form as √466.
Estimating √466 by hand
√466 falls between √441 = 21 and √484 = 22, so it starts at 21.something. The Babylonian method refines that guess by averaging it with 466 ÷ the guess:
| Step | Estimate |
| Start | 21 |
| Step 1 | 21.595238 |
| Step 2 | 21.587035 |
| Step 3 | 21.587033 |
After 3 steps the estimate is 21.587033, matching √466 ≈ 21.587033.
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Frequently asked questions
Is 466 a perfect square?
No, 466 is not a perfect square, so its square root is irrational and approximately 21.587033.
What is the square root of 466 in simplest radical form?
√466 = √466 in simplest radical form.