Square Root Calculator › Square root of 423
Square Root of 423
√423 ≈ 20.566964
423 is not a perfect square, so √423 is an irrational number that is approximately 20.566964. In simplified radical form, √423 = 3√47.
√423 lies between √400 = 20 and √441 = 21.
| Number | 423 |
| Square root (decimal) | 20.566964 |
| Simplified radical | 3√47 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 423
423 = 3^2 × 47. This is what determines the simplified radical: for √423 you pull out one copy of any prime that appears twice.
Here the paired factor 3 comes out from under the root, leaving √423 = 3√47.
Estimating √423 by hand
√423 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 423 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.575 |
| Step 2 | 20.566965 |
| Step 3 | 20.566964 |
After 3 steps the estimate is 20.566964, matching √423 ≈ 20.566964.
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Frequently asked questions
Is 423 a perfect square?
No, 423 is not a perfect square, so its square root is irrational and approximately 20.566964.
What is the square root of 423 in simplest radical form?
√423 = 3√47 in simplest radical form.