Square Root Calculator › Square root of 414
Square Root of 414
√414 ≈ 20.34699
414 is not a perfect square, so √414 is an irrational number that is approximately 20.34699. In simplified radical form, √414 = 3√46.
√414 lies between √400 = 20 and √441 = 21.
| Number | 414 |
| Square root (decimal) | 20.34699 |
| Simplified radical | 3√46 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 414
414 = 2 × 3^2 × 23. This is what determines the simplified radical: for √414 you pull out one copy of any prime that appears twice.
Here the paired factor 3 comes out from under the root, leaving √414 = 3√46.
Estimating √414 by hand
√414 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 414 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.35 |
| Step 2 | 20.34699 |
| Step 3 | 20.34699 |
After 3 steps the estimate is 20.34699, matching √414 ≈ 20.34699.
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Frequently asked questions
Is 414 a perfect square?
No, 414 is not a perfect square, so its square root is irrational and approximately 20.34699.
What is the square root of 414 in simplest radical form?
√414 = 3√46 in simplest radical form.