Square Root Calculator › Square root of 417
Square Root of 417
√417 ≈ 20.420578
417 is not a perfect square, so √417 is an irrational number that is approximately 20.420578. In simplified radical form, √417 = √417.
√417 lies between √400 = 20 and √441 = 21.
| Number | 417 |
| Square root (decimal) | 20.420578 |
| Simplified radical | √417 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 417
417 = 3 × 139. This is what determines the simplified radical: for √417 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √417 is already in simplest radical form as √417.
Estimating √417 by hand
√417 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 417 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.425 |
| Step 2 | 20.420578 |
| Step 3 | 20.420578 |
After 3 steps the estimate is 20.420578, matching √417 ≈ 20.420578.
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Frequently asked questions
Is 417 a perfect square?
No, 417 is not a perfect square, so its square root is irrational and approximately 20.420578.
What is the square root of 417 in simplest radical form?
√417 = √417 in simplest radical form.