Square Root of 76
The square root of 76 is approximately 8.717798, or exactly 2√19 in simplified radical form. No, 76 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √76 sits
8² = 64 and 9² = 81, and 76 falls between them, so √76 is between 8 and 9. It's closer to 9 (12 above 64, 5 below 81), which matches the decimal 8.717798.
| k | k² | Compared to 76 |
|---|---|---|
| 6 | 36 | 40 below |
| 7 | 49 | 27 below |
| 8 | 64 | 12 below |
| 9 | 81 | 5 above |
| 10 | 100 | 24 above |
| 11 | 121 | 45 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 8), divide 76 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 76 ÷ x | New guess (x + 76÷x) ÷ 2 |
|---|---|---|---|
| 1 | 8 | 9.5 | 8.75 |
| 2 | 8.75 | 8.685714 | 8.717857 |
| 3 | 8.717857 | 8.717739 | 8.717798 |
| 4 | 8.717798 | 8.717798 | 8.717798 |
After 4 passes the guess is 8.717798, already matching √76 = 8.717798 to six decimal places.
Exact form from the prime factors
76 = 2^2 × 19. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √76 = 2√19. That's the exact value; the decimal 8.717798 is a rounding of it.
Frequently Asked Questions
What is the square root of 76?
The square root of 76 is approximately 8.717798, or exactly 2√19 in simplified radical form.
Is 76 a perfect square?
No, 76 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √76 between?
Between 8 and 9, because 8² = 64 and 9² = 81.
What are the square roots of the numbers right next to 76?
√75 is about 8.6603, and √77 is about 8.775. 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 √76 squared?
Exactly 76 -- squaring undoes the square root. Squaring the rounded decimal 8.717798 gives 76, which is why the radical form 2√19 is the exact answer and the decimal is an approximation.
What is the negative square root of 76?
−8.717798. Every positive number has two square roots, one positive and one negative, because (−8.717798)² is also 76. The radical sign √ by convention means the positive one.
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