Square Root of 77
The square root of 77 is approximately 8.774964, or exactly √77 in simplified radical form. No, 77 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √77 sits
8² = 64 and 9² = 81, and 77 falls between them, so √77 is between 8 and 9. It's closer to 9 (13 above 64, 4 below 81), which matches the decimal 8.774964.
| k | k² | Compared to 77 |
|---|---|---|
| 6 | 36 | 41 below |
| 7 | 49 | 28 below |
| 8 | 64 | 13 below |
| 9 | 81 | 4 above |
| 10 | 100 | 23 above |
| 11 | 121 | 44 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 8), divide 77 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 77 ÷ x | New guess (x + 77÷x) ÷ 2 |
|---|---|---|---|
| 1 | 8 | 9.625 | 8.8125 |
| 2 | 8.8125 | 8.737589 | 8.775044 |
| 3 | 8.775044 | 8.774884 | 8.774964 |
| 4 | 8.774964 | 8.774964 | 8.774964 |
After 4 passes the guess is 8.774964, already matching √77 = 8.774964 to six decimal places.
Exact form from the prime factors
77 = 7 × 11. No prime appears twice, so nothing can be pulled out of the root -- √77 is already in simplest form.
Frequently Asked Questions
What is the square root of 77?
The square root of 77 is approximately 8.774964, or exactly √77 in simplified radical form.
Is 77 a perfect square?
No, 77 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √77 between?
Between 8 and 9, because 8² = 64 and 9² = 81.
What are the square roots of the numbers right next to 77?
√76 is about 8.7178, and √78 is about 8.8318. Square roots increase more slowly as the number gets bigger, which is why consecutive whole numbers have roots that get closer together the higher you go.
What is √77 squared?
Exactly 77 -- squaring undoes the square root. Squaring the rounded decimal 8.774964 gives 77, which is why the radical form √77 is the exact answer and the decimal is an approximation.
What is the negative square root of 77?
−8.774964. Every positive number has two square roots, one positive and one negative, because (−8.774964)² is also 77. The radical sign √ by convention means the positive one.
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