Square Root Calculator › Square root of 408
Square Root of 408
√408 ≈ 20.19901
408 is not a perfect square, so √408 is an irrational number that is approximately 20.19901. In simplified radical form, √408 = 2√102.
√408 lies between √400 = 20 and √441 = 21.
| Number | 408 |
| Square root (decimal) | 20.19901 |
| Simplified radical | 2√102 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 408
408 = 2^3 × 3 × 17. This is what determines the simplified radical: for √408 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √408 = 2√102.
Estimating √408 by hand
√408 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 408 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.2 |
| Step 2 | 20.19901 |
| Step 3 | 20.19901 |
After 3 steps the estimate is 20.19901, matching √408 ≈ 20.19901.
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Frequently asked questions
Is 408 a perfect square?
No, 408 is not a perfect square, so its square root is irrational and approximately 20.19901.
What is the square root of 408 in simplest radical form?
√408 = 2√102 in simplest radical form.