Square Root of 82
The square root of 82 is approximately 9.055385, or exactly √82 in simplified radical form. No, 82 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √82 sits
9² = 81 and 10² = 100, and 82 falls between them, so √82 is between 9 and 10. It's closer to 9 (1 above 81, 18 below 100), which matches the decimal 9.055385.
| k | k² | Compared to 82 |
|---|---|---|
| 7 | 49 | 33 below |
| 8 | 64 | 18 below |
| 9 | 81 | 1 below |
| 10 | 100 | 18 above |
| 11 | 121 | 39 above |
| 12 | 144 | 62 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 9), divide 82 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 82 ÷ x | New guess (x + 82÷x) ÷ 2 |
|---|---|---|---|
| 1 | 9 | 9.111111 | 9.055556 |
| 2 | 9.055556 | 9.055215 | 9.055385 |
| 3 | 9.055385 | 9.055385 | 9.055385 |
| 4 | 9.055385 | 9.055385 | 9.055385 |
After 4 passes the guess is 9.055385, already matching √82 = 9.055385 to six decimal places.
Exact form from the prime factors
82 = 2 × 41. No prime appears twice, so nothing can be pulled out of the root -- √82 is already in simplest form.
Frequently Asked Questions
What is the square root of 82?
The square root of 82 is approximately 9.055385, or exactly √82 in simplified radical form.
Is 82 a perfect square?
No, 82 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √82 between?
Between 9 and 10, because 9² = 81 and 10² = 100.
What are the square roots of the numbers right next to 82?
√81 is exactly 9, and √83 is about 9.1104. 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 √82 squared?
Exactly 82 -- squaring undoes the square root. Squaring the rounded decimal 9.055385 gives 82, which is why the radical form √82 is the exact answer and the decimal is an approximation.
What is the negative square root of 82?
−9.055385. Every positive number has two square roots, one positive and one negative, because (−9.055385)² is also 82. The radical sign √ by convention means the positive one.
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