Square Root Calculator › Square root of 458
Square Root of 458
√458 ≈ 21.400935
458 is not a perfect square, so √458 is an irrational number that is approximately 21.400935. In simplified radical form, √458 = √458.
√458 lies between √441 = 21 and √484 = 22.
| Number | 458 |
| Square root (decimal) | 21.400935 |
| Simplified radical | √458 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 458
458 = 2 × 229. This is what determines the simplified radical: for √458 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √458 is already in simplest radical form as √458.
Estimating √458 by hand
√458 falls between √441 = 21 and √484 = 22, so it starts at 21.something. The Babylonian method refines that guess by averaging it with 458 ÷ the guess:
| Step | Estimate |
| Start | 21 |
| Step 1 | 21.404762 |
| Step 2 | 21.400935 |
| Step 3 | 21.400935 |
After 3 steps the estimate is 21.400935, matching √458 ≈ 21.400935.
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Frequently asked questions
Is 458 a perfect square?
No, 458 is not a perfect square, so its square root is irrational and approximately 21.400935.
What is the square root of 458 in simplest radical form?
√458 = √458 in simplest radical form.