Square Root Calculator › Square root of 412
Square Root of 412
√412 ≈ 20.297783
412 is not a perfect square, so √412 is an irrational number that is approximately 20.297783. In simplified radical form, √412 = 2√103.
√412 lies between √400 = 20 and √441 = 21.
| Number | 412 |
| Square root (decimal) | 20.297783 |
| Simplified radical | 2√103 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 412
412 = 2^2 × 103. This is what determines the simplified radical: for √412 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √412 = 2√103.
Estimating √412 by hand
√412 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 412 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.3 |
| Step 2 | 20.297783 |
| Step 3 | 20.297783 |
After 3 steps the estimate is 20.297783, matching √412 ≈ 20.297783.
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Frequently asked questions
Is 412 a perfect square?
No, 412 is not a perfect square, so its square root is irrational and approximately 20.297783.
What is the square root of 412 in simplest radical form?
√412 = 2√103 in simplest radical form.