Square Root Calculator › Square root of 421
Square Root of 421
√421 ≈ 20.518285
421 is not a perfect square, so √421 is an irrational number that is approximately 20.518285. In simplified radical form, √421 = √421.
√421 lies between √400 = 20 and √441 = 21.
| Number | 421 |
| Square root (decimal) | 20.518285 |
| Simplified radical | √421 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 421
421 = 421. This is what determines the simplified radical: for √421 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √421 is already in simplest radical form as √421.
Estimating √421 by hand
√421 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 421 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.525 |
| Step 2 | 20.518286 |
| Step 3 | 20.518285 |
After 3 steps the estimate is 20.518285, matching √421 ≈ 20.518285.
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Frequently asked questions
Is 421 a perfect square?
No, 421 is not a perfect square, so its square root is irrational and approximately 20.518285.
What is the square root of 421 in simplest radical form?
√421 = √421 in simplest radical form.