Square Root of 85
The square root of 85 is approximately 9.219544, or exactly √85 in simplified radical form. No, 85 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √85 sits
9² = 81 and 10² = 100, and 85 falls between them, so √85 is between 9 and 10. It's closer to 9 (4 above 81, 15 below 100), which matches the decimal 9.219544.
| k | k² | Compared to 85 |
|---|---|---|
| 7 | 49 | 36 below |
| 8 | 64 | 21 below |
| 9 | 81 | 4 below |
| 10 | 100 | 15 above |
| 11 | 121 | 36 above |
| 12 | 144 | 59 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 9), divide 85 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 85 ÷ x | New guess (x + 85÷x) ÷ 2 |
|---|---|---|---|
| 1 | 9 | 9.444444 | 9.222222 |
| 2 | 9.222222 | 9.216867 | 9.219545 |
| 3 | 9.219545 | 9.219544 | 9.219544 |
| 4 | 9.219544 | 9.219544 | 9.219544 |
After 4 passes the guess is 9.219544, already matching √85 = 9.219544 to six decimal places.
Exact form from the prime factors
85 = 5 × 17. No prime appears twice, so nothing can be pulled out of the root -- √85 is already in simplest form.
Frequently Asked Questions
What is the square root of 85?
The square root of 85 is approximately 9.219544, or exactly √85 in simplified radical form.
Is 85 a perfect square?
No, 85 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √85 between?
Between 9 and 10, because 9² = 81 and 10² = 100.
What are the square roots of the numbers right next to 85?
√84 is about 9.1652, and √86 is about 9.2736. 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 √85 squared?
Exactly 85 -- squaring undoes the square root. Squaring the rounded decimal 9.219544 gives 85, which is why the radical form √85 is the exact answer and the decimal is an approximation.
What is the negative square root of 85?
−9.219544. Every positive number has two square roots, one positive and one negative, because (−9.219544)² is also 85. The radical sign √ by convention means the positive one.
More Number Theory
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