Square Root Calculator › Square root of 406
Square Root of 406
√406 ≈ 20.149442
406 is not a perfect square, so √406 is an irrational number that is approximately 20.149442. In simplified radical form, √406 = √406.
√406 lies between √400 = 20 and √441 = 21.
| Number | 406 |
| Square root (decimal) | 20.149442 |
| Simplified radical | √406 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 406
406 = 2 × 7 × 29. This is what determines the simplified radical: for √406 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √406 is already in simplest radical form as √406.
Estimating √406 by hand
√406 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 406 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.15 |
| Step 2 | 20.149442 |
| Step 3 | 20.149442 |
After 3 steps the estimate is 20.149442, matching √406 ≈ 20.149442.
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Frequently asked questions
Is 406 a perfect square?
No, 406 is not a perfect square, so its square root is irrational and approximately 20.149442.
What is the square root of 406 in simplest radical form?
√406 = √406 in simplest radical form.