Square Root Calculator › Square root of 422
Square Root of 422
√422 ≈ 20.542639
422 is not a perfect square, so √422 is an irrational number that is approximately 20.542639. In simplified radical form, √422 = √422.
√422 lies between √400 = 20 and √441 = 21.
| Number | 422 |
| Square root (decimal) | 20.542639 |
| Simplified radical | √422 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 422
422 = 2 × 211. This is what determines the simplified radical: for √422 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √422 is already in simplest radical form as √422.
Estimating √422 by hand
√422 falls between √400 = 20 and √441 = 21, so it starts at 20.something. The Babylonian method refines that guess by averaging it with 422 ÷ the guess:
| Step | Estimate |
| Start | 20 |
| Step 1 | 20.55 |
| Step 2 | 20.54264 |
| Step 3 | 20.542639 |
After 3 steps the estimate is 20.542639, matching √422 ≈ 20.542639.
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Frequently asked questions
Is 422 a perfect square?
No, 422 is not a perfect square, so its square root is irrational and approximately 20.542639.
What is the square root of 422 in simplest radical form?
√422 = √422 in simplest radical form.