Square Root Calculator › Square root of 451
Square Root of 451
√451 ≈ 21.236761
451 is not a perfect square, so √451 is an irrational number that is approximately 21.236761. In simplified radical form, √451 = √451.
√451 lies between √441 = 21 and √484 = 22.
| Number | 451 |
| Square root (decimal) | 21.236761 |
| Simplified radical | √451 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 451
451 = 11 × 41. This is what determines the simplified radical: for √451 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √451 is already in simplest radical form as √451.
Estimating √451 by hand
√451 falls between √441 = 21 and √484 = 22, so it starts at 21.something. The Babylonian method refines that guess by averaging it with 451 ÷ the guess:
| Step | Estimate |
| Start | 21 |
| Step 1 | 21.238095 |
| Step 2 | 21.236761 |
| Step 3 | 21.236761 |
After 3 steps the estimate is 21.236761, matching √451 ≈ 21.236761.
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Frequently asked questions
Is 451 a perfect square?
No, 451 is not a perfect square, so its square root is irrational and approximately 21.236761.
What is the square root of 451 in simplest radical form?
√451 = √451 in simplest radical form.