Square Root Calculator › Square root of 416
Square Root of 416
√416 ≈ 20.396078
416 is not a perfect square, so √416 is an irrational number that is approximately 20.396078. In simplified radical form, √416 = 4√26.
√416 lies between √400 = 20 and √441 = 21.
| Number | 416 |
| Square root (decimal) | 20.396078 |
| Simplified radical | 4√26 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 416
416 = 2^5 × 13. This is what determines the simplified radical: for √416 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √416 = 4√26.
Estimating √416 by hand
√416 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 416 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.4 |
| Step 2 | 20.396078 |
| Step 3 | 20.396078 |
After 3 steps the estimate is 20.396078, matching √416 ≈ 20.396078.
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Frequently asked questions
Is 416 a perfect square?
No, 416 is not a perfect square, so its square root is irrational and approximately 20.396078.
What is the square root of 416 in simplest radical form?
√416 = 4√26 in simplest radical form.