Square Root Calculator › Square root of 82
Square Root of 82
√82 ≈ 9.055385
82 is not a perfect square, so √82 is an irrational number that is approximately 9.055385. In simplified radical form, √82 = √82.
√82 lies between √81 = 9 and √100 = 10.
| Number | 82 |
| Square root (decimal) | 9.055385 |
| Simplified radical | √82 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 82
82 = 2 × 41. This is what determines the simplified radical: for √82 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √82 is already in simplest radical form as √82.
Estimating √82 by hand
√82 falls between √81 = 9 and √100 = 10, so it starts at 9.something. The Babylonian method refines that guess by averaging it with 82 ÷ the guess:
| Step | Estimate |
| Start | 9 |
| Step 1 | 9.055556 |
| Step 2 | 9.055385 |
| Step 3 | 9.055385 |
After 3 steps the estimate is 9.055385, matching √82 ≈ 9.055385.
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Frequently asked questions
Is 82 a perfect square?
No, 82 is not a perfect square, so its square root is irrational and approximately 9.055385.
What is the square root of 82 in simplest radical form?
√82 = √82 in simplest radical form.