Square Root of 89
The square root of 89 is approximately 9.433981, or exactly √89 in simplified radical form. No, 89 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √89 sits
9² = 81 and 10² = 100, and 89 falls between them, so √89 is between 9 and 10. It's closer to 9 (8 above 81, 11 below 100), which matches the decimal 9.433981.
| k | k² | Compared to 89 |
|---|---|---|
| 7 | 49 | 40 below |
| 8 | 64 | 25 below |
| 9 | 81 | 8 below |
| 10 | 100 | 11 above |
| 11 | 121 | 32 above |
| 12 | 144 | 55 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 9), divide 89 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 89 ÷ x | New guess (x + 89÷x) ÷ 2 |
|---|---|---|---|
| 1 | 9 | 9.888889 | 9.444444 |
| 2 | 9.444444 | 9.423529 | 9.433987 |
| 3 | 9.433987 | 9.433975 | 9.433981 |
| 4 | 9.433981 | 9.433981 | 9.433981 |
After 4 passes the guess is 9.433981, already matching √89 = 9.433981 to six decimal places.
Exact form from the prime factors
89 = 89. No prime appears twice, so nothing can be pulled out of the root -- √89 is already in simplest form.
Frequently Asked Questions
What is the square root of 89?
The square root of 89 is approximately 9.433981, or exactly √89 in simplified radical form.
Is 89 a perfect square?
No, 89 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √89 between?
Between 9 and 10, because 9² = 81 and 10² = 100.
What are the square roots of the numbers right next to 89?
√88 is about 9.3808, and √90 is about 9.4868. 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 √89 squared?
Exactly 89 -- squaring undoes the square root. Squaring the rounded decimal 9.433981 gives 89, which is why the radical form √89 is the exact answer and the decimal is an approximation.
What is the negative square root of 89?
−9.433981. Every positive number has two square roots, one positive and one negative, because (−9.433981)² is also 89. The radical sign √ by convention means the positive one.
More Number Theory
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