Square Root of 86
The square root of 86 is approximately 9.273618, or exactly √86 in simplified radical form. No, 86 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √86 sits
9² = 81 and 10² = 100, and 86 falls between them, so √86 is between 9 and 10. It's closer to 9 (5 above 81, 14 below 100), which matches the decimal 9.273618.
| k | k² | Compared to 86 |
|---|---|---|
| 7 | 49 | 37 below |
| 8 | 64 | 22 below |
| 9 | 81 | 5 below |
| 10 | 100 | 14 above |
| 11 | 121 | 35 above |
| 12 | 144 | 58 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 9), divide 86 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 86 ÷ x | New guess (x + 86÷x) ÷ 2 |
|---|---|---|---|
| 1 | 9 | 9.555556 | 9.277778 |
| 2 | 9.277778 | 9.269461 | 9.273619 |
| 3 | 9.273619 | 9.273618 | 9.273618 |
| 4 | 9.273618 | 9.273618 | 9.273618 |
After 4 passes the guess is 9.273618, already matching √86 = 9.273618 to six decimal places.
Exact form from the prime factors
86 = 2 × 43. No prime appears twice, so nothing can be pulled out of the root -- √86 is already in simplest form.
Frequently Asked Questions
What is the square root of 86?
The square root of 86 is approximately 9.273618, or exactly √86 in simplified radical form.
Is 86 a perfect square?
No, 86 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √86 between?
Between 9 and 10, because 9² = 81 and 10² = 100.
What are the square roots of the numbers right next to 86?
√85 is about 9.2195, and √87 is about 9.3274. 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 √86 squared?
Exactly 86 -- squaring undoes the square root. Squaring the rounded decimal 9.273618 gives 86, which is why the radical form √86 is the exact answer and the decimal is an approximation.
What is the negative square root of 86?
−9.273618. Every positive number has two square roots, one positive and one negative, because (−9.273618)² is also 86. The radical sign √ by convention means the positive one.
More Number Theory
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