Square Root Calculator › Square root of 427
Square Root of 427
√427 ≈ 20.663978
427 is not a perfect square, so √427 is an irrational number that is approximately 20.663978. In simplified radical form, √427 = √427.
√427 lies between √400 = 20 and √441 = 21.
| Number | 427 |
| Square root (decimal) | 20.663978 |
| Simplified radical | √427 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 427
427 = 7 × 61. This is what determines the simplified radical: for √427 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √427 is already in simplest radical form as √427.
Estimating √427 by hand
√427 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 427 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.675 |
| Step 2 | 20.663981 |
| Step 3 | 20.663978 |
After 3 steps the estimate is 20.663978, matching √427 ≈ 20.663978.
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Frequently asked questions
Is 427 a perfect square?
No, 427 is not a perfect square, so its square root is irrational and approximately 20.663978.
What is the square root of 427 in simplest radical form?
√427 = √427 in simplest radical form.