Square Root Calculator › Square root of 65
Square Root of 65
√65 ≈ 8.062258
65 is not a perfect square, so √65 is an irrational number that is approximately 8.062258. In simplified radical form, √65 = √65.
√65 lies between √64 = 8 and √81 = 9.
| Number | 65 |
| Square root (decimal) | 8.062258 |
| Simplified radical | √65 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 65
65 = 5 × 13. This is what determines the simplified radical: for √65 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √65 is already in simplest radical form as √65.
Estimating √65 by hand
√65 falls between √64 = 8 and √81 = 9, so it starts at 8.something. The Babylonian method refines that guess by averaging it with 65 ÷ the guess:
| Step | Estimate |
| Start | 8 |
| Step 1 | 8.0625 |
| Step 2 | 8.062258 |
| Step 3 | 8.062258 |
After 3 steps the estimate is 8.062258, matching √65 ≈ 8.062258.
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Frequently asked questions
Is 65 a perfect square?
No, 65 is not a perfect square, so its square root is irrational and approximately 8.062258.
What is the square root of 65 in simplest radical form?
√65 = √65 in simplest radical form.