Square Root Calculator › Square root of 405
Square Root of 405
√405 ≈ 20.124612
405 is not a perfect square, so √405 is an irrational number that is approximately 20.124612. In simplified radical form, √405 = 9√5.
√405 lies between √400 = 20 and √441 = 21.
| Number | 405 |
| Square root (decimal) | 20.124612 |
| Simplified radical | 9√5 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 405
405 = 3^4 × 5. This is what determines the simplified radical: for √405 you pull out one copy of any prime that appears twice.
Here the paired factor 3 comes out from under the root, leaving √405 = 9√5.
Estimating √405 by hand
√405 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 405 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.125 |
| Step 2 | 20.124612 |
| Step 3 | 20.124612 |
After 3 steps the estimate is 20.124612, matching √405 ≈ 20.124612.
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Frequently asked questions
Is 405 a perfect square?
No, 405 is not a perfect square, so its square root is irrational and approximately 20.124612.
What is the square root of 405 in simplest radical form?
√405 = 9√5 in simplest radical form.