Square Root Calculator › Square root of 429
Square Root of 429
√429 ≈ 20.712315
429 is not a perfect square, so √429 is an irrational number that is approximately 20.712315. In simplified radical form, √429 = √429.
√429 lies between √400 = 20 and √441 = 21.
| Number | 429 |
| Square root (decimal) | 20.712315 |
| Simplified radical | √429 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 429
429 = 3 × 11 × 13. This is what determines the simplified radical: for √429 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √429 is already in simplest radical form as √429.
Estimating √429 by hand
√429 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 429 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.725 |
| Step 2 | 20.712319 |
| Step 3 | 20.712315 |
After 3 steps the estimate is 20.712315, matching √429 ≈ 20.712315.
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Frequently asked questions
Is 429 a perfect square?
No, 429 is not a perfect square, so its square root is irrational and approximately 20.712315.
What is the square root of 429 in simplest radical form?
√429 = √429 in simplest radical form.