Square Root Calculator › Square root of 407
Square Root of 407
√407 ≈ 20.174241
407 is not a perfect square, so √407 is an irrational number that is approximately 20.174241. In simplified radical form, √407 = √407.
√407 lies between √400 = 20 and √441 = 21.
| Number | 407 |
| Square root (decimal) | 20.174241 |
| Simplified radical | √407 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 407
407 = 11 × 37. This is what determines the simplified radical: for √407 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √407 is already in simplest radical form as √407.
Estimating √407 by hand
√407 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 407 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.175 |
| Step 2 | 20.174241 |
| Step 3 | 20.174241 |
After 3 steps the estimate is 20.174241, matching √407 ≈ 20.174241.
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Frequently asked questions
Is 407 a perfect square?
No, 407 is not a perfect square, so its square root is irrational and approximately 20.174241.
What is the square root of 407 in simplest radical form?
√407 = √407 in simplest radical form.