Square Root Calculator › Square root of 77
Square Root of 77
√77 ≈ 8.774964
77 is not a perfect square, so √77 is an irrational number that is approximately 8.774964. In simplified radical form, √77 = √77.
√77 lies between √64 = 8 and √81 = 9.
| Number | 77 |
| Square root (decimal) | 8.774964 |
| Simplified radical | √77 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 77
77 = 7 × 11. This is what determines the simplified radical: for √77 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √77 is already in simplest radical form as √77.
Estimating √77 by hand
√77 falls between √64 = 8 and √81 = 9, so it starts at 8.something. The Babylonian method refines that guess by averaging it with 77 ÷ the guess:
| Step | Estimate |
| Start | 8 |
| Step 1 | 8.8125 |
| Step 2 | 8.775044 |
| Step 3 | 8.774964 |
After 3 steps the estimate is 8.774964, matching √77 ≈ 8.774964.
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Frequently asked questions
Is 77 a perfect square?
No, 77 is not a perfect square, so its square root is irrational and approximately 8.774964.
What is the square root of 77 in simplest radical form?
√77 = √77 in simplest radical form.