Square Root Calculator › Square root of 78
Square Root of 78
√78 ≈ 8.831761
78 is not a perfect square, so √78 is an irrational number that is approximately 8.831761. In simplified radical form, √78 = √78.
√78 lies between √64 = 8 and √81 = 9.
| Number | 78 |
| Square root (decimal) | 8.831761 |
| Simplified radical | √78 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 78
78 = 2 × 3 × 13. This is what determines the simplified radical: for √78 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √78 is already in simplest radical form as √78.
Estimating √78 by hand
√78 falls between √64 = 8 and √81 = 9, so it starts at 8.something. The Babylonian method refines that guess by averaging it with 78 ÷ the guess:
| Step | Estimate |
| Start | 8 |
| Step 1 | 8.875 |
| Step 2 | 8.831866 |
| Step 3 | 8.831761 |
After 3 steps the estimate is 8.831761, matching √78 ≈ 8.831761.
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Frequently asked questions
Is 78 a perfect square?
No, 78 is not a perfect square, so its square root is irrational and approximately 8.831761.
What is the square root of 78 in simplest radical form?
√78 = √78 in simplest radical form.