Square Root Calculator › Square root of 415
Square Root of 415
√415 ≈ 20.371549
415 is not a perfect square, so √415 is an irrational number that is approximately 20.371549. In simplified radical form, √415 = √415.
√415 lies between √400 = 20 and √441 = 21.
| Number | 415 |
| Square root (decimal) | 20.371549 |
| Simplified radical | √415 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 415
415 = 5 × 83. This is what determines the simplified radical: for √415 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √415 is already in simplest radical form as √415.
Estimating √415 by hand
√415 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 415 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.375 |
| Step 2 | 20.371549 |
| Step 3 | 20.371549 |
After 3 steps the estimate is 20.371549, matching √415 ≈ 20.371549.
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Frequently asked questions
Is 415 a perfect square?
No, 415 is not a perfect square, so its square root is irrational and approximately 20.371549.
What is the square root of 415 in simplest radical form?
√415 = √415 in simplest radical form.