Square Root Calculator › Square root of 409
Square Root of 409
√409 ≈ 20.223748
409 is not a perfect square, so √409 is an irrational number that is approximately 20.223748. In simplified radical form, √409 = √409.
√409 lies between √400 = 20 and √441 = 21.
| Number | 409 |
| Square root (decimal) | 20.223748 |
| Simplified radical | √409 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 409
409 = 409. This is what determines the simplified radical: for √409 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √409 is already in simplest radical form as √409.
Estimating √409 by hand
√409 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 409 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.225 |
| Step 2 | 20.223748 |
| Step 3 | 20.223748 |
After 3 steps the estimate is 20.223748, matching √409 ≈ 20.223748.
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Frequently asked questions
Is 409 a perfect square?
No, 409 is not a perfect square, so its square root is irrational and approximately 20.223748.
What is the square root of 409 in simplest radical form?
√409 = √409 in simplest radical form.