Square Root Calculator › Square root of 212
Square Root of 212
√212 ≈ 14.56022
212 is not a perfect square, so √212 is an irrational number that is approximately 14.56022. In simplified radical form, √212 = 2√53.
√212 lies between √196 = 14 and √225 = 15.
| Number | 212 |
| Square root (decimal) | 14.56022 |
| Simplified radical | 2√53 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 212
212 = 2^2 × 53. This is what determines the simplified radical: for √212 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √212 = 2√53.
Estimating √212 by hand
√212 falls between √196 = 14 and √225 = 15, so it starts at 14.something. The Babylonian method refines that guess by averaging it with 212 ÷ the guess:
| Step | Estimate |
| Start | 14 |
| Step 1 | 14.571429 |
| Step 2 | 14.560224 |
| Step 3 | 14.56022 |
After 3 steps the estimate is 14.56022, matching √212 ≈ 14.56022.
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Frequently asked questions
Is 212 a perfect square?
No, 212 is not a perfect square, so its square root is irrational and approximately 14.56022.
What is the square root of 212 in simplest radical form?
√212 = 2√53 in simplest radical form.