Square Root Calculator › Square root of 213
Square Root of 213
√213 ≈ 14.59452
213 is not a perfect square, so √213 is an irrational number that is approximately 14.59452. In simplified radical form, √213 = √213.
√213 lies between √196 = 14 and √225 = 15.
| Number | 213 |
| Square root (decimal) | 14.59452 |
| Simplified radical | √213 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 213
213 = 3 × 71. This is what determines the simplified radical: for √213 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √213 is already in simplest radical form as √213.
Estimating √213 by hand
√213 falls between √196 = 14 and √225 = 15, so it starts at 14.something. The Babylonian method refines that guess by averaging it with 213 ÷ the guess:
| Step | Estimate |
| Start | 14 |
| Step 1 | 14.607143 |
| Step 2 | 14.594525 |
| Step 3 | 14.59452 |
After 3 steps the estimate is 14.59452, matching √213 ≈ 14.59452.
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Frequently asked questions
Is 213 a perfect square?
No, 213 is not a perfect square, so its square root is irrational and approximately 14.59452.
What is the square root of 213 in simplest radical form?
√213 = √213 in simplest radical form.