Square Root Calculator › Square root of 404
Square Root of 404
√404 ≈ 20.099751
404 is not a perfect square, so √404 is an irrational number that is approximately 20.099751. In simplified radical form, √404 = 2√101.
√404 lies between √400 = 20 and √441 = 21.
| Number | 404 |
| Square root (decimal) | 20.099751 |
| Simplified radical | 2√101 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 404
404 = 2^2 × 101. This is what determines the simplified radical: for √404 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √404 = 2√101.
Estimating √404 by hand
√404 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 404 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.1 |
| Step 2 | 20.099751 |
| Step 3 | 20.099751 |
After 3 steps the estimate is 20.099751, matching √404 ≈ 20.099751.
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Frequently asked questions
Is 404 a perfect square?
No, 404 is not a perfect square, so its square root is irrational and approximately 20.099751.
What is the square root of 404 in simplest radical form?
√404 = 2√101 in simplest radical form.