Square Root of 78
The square root of 78 is approximately 8.831761, or exactly √78 in simplified radical form. No, 78 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √78 sits
8² = 64 and 9² = 81, and 78 falls between them, so √78 is between 8 and 9. It's closer to 9 (14 above 64, 3 below 81), which matches the decimal 8.831761.
| k | k² | Compared to 78 |
|---|---|---|
| 6 | 36 | 42 below |
| 7 | 49 | 29 below |
| 8 | 64 | 14 below |
| 9 | 81 | 3 above |
| 10 | 100 | 22 above |
| 11 | 121 | 43 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 8), divide 78 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 78 ÷ x | New guess (x + 78÷x) ÷ 2 |
|---|---|---|---|
| 1 | 8 | 9.75 | 8.875 |
| 2 | 8.875 | 8.788732 | 8.831866 |
| 3 | 8.831866 | 8.831656 | 8.831761 |
| 4 | 8.831761 | 8.831761 | 8.831761 |
After 4 passes the guess is 8.831761, already matching √78 = 8.831761 to six decimal places.
Exact form from the prime factors
78 = 2 × 3 × 13. No prime appears twice, so nothing can be pulled out of the root -- √78 is already in simplest form.
Frequently Asked Questions
What is the square root of 78?
The square root of 78 is approximately 8.831761, or exactly √78 in simplified radical form.
Is 78 a perfect square?
No, 78 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √78 between?
Between 8 and 9, because 8² = 64 and 9² = 81.
What are the square roots of the numbers right next to 78?
√77 is about 8.775, and √79 is about 8.8882. 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 √78 squared?
Exactly 78 -- squaring undoes the square root. Squaring the rounded decimal 8.831761 gives 78, which is why the radical form √78 is the exact answer and the decimal is an approximation.
What is the negative square root of 78?
−8.831761. Every positive number has two square roots, one positive and one negative, because (−8.831761)² is also 78. The radical sign √ by convention means the positive one.
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