Square Root Calculator › Square root of 401
Square Root of 401
√401 ≈ 20.024984
401 is not a perfect square, so √401 is an irrational number that is approximately 20.024984. In simplified radical form, √401 = √401.
√401 lies between √400 = 20 and √441 = 21.
| Number | 401 |
| Square root (decimal) | 20.024984 |
| Simplified radical | √401 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 401
401 = 401. This is what determines the simplified radical: for √401 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √401 is already in simplest radical form as √401.
Estimating √401 by hand
√401 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 401 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.025 |
| Step 2 | 20.024984 |
| Step 3 | 20.024984 |
After 3 steps the estimate is 20.024984, matching √401 ≈ 20.024984.
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Frequently asked questions
Is 401 a perfect square?
No, 401 is not a perfect square, so its square root is irrational and approximately 20.024984.
What is the square root of 401 in simplest radical form?
√401 = √401 in simplest radical form.