Square Root Calculator › Square root of 413
Square Root of 413
√413 ≈ 20.322401
413 is not a perfect square, so √413 is an irrational number that is approximately 20.322401. In simplified radical form, √413 = √413.
√413 lies between √400 = 20 and √441 = 21.
| Number | 413 |
| Square root (decimal) | 20.322401 |
| Simplified radical | √413 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 413
413 = 7 × 59. This is what determines the simplified radical: for √413 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √413 is already in simplest radical form as √413.
Estimating √413 by hand
√413 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 413 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.325 |
| Step 2 | 20.322402 |
| Step 3 | 20.322401 |
After 3 steps the estimate is 20.322401, matching √413 ≈ 20.322401.
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Frequently asked questions
Is 413 a perfect square?
No, 413 is not a perfect square, so its square root is irrational and approximately 20.322401.
What is the square root of 413 in simplest radical form?
√413 = √413 in simplest radical form.