Square Root of 91
The square root of 91 is approximately 9.539392, or exactly √91 in simplified radical form. No, 91 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √91 sits
9² = 81 and 10² = 100, and 91 falls between them, so √91 is between 9 and 10. It's closer to 10 (10 above 81, 9 below 100), which matches the decimal 9.539392.
| k | k² | Compared to 91 |
|---|---|---|
| 7 | 49 | 42 below |
| 8 | 64 | 27 below |
| 9 | 81 | 10 below |
| 10 | 100 | 9 above |
| 11 | 121 | 30 above |
| 12 | 144 | 53 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 9), divide 91 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 91 ÷ x | New guess (x + 91÷x) ÷ 2 |
|---|---|---|---|
| 1 | 9 | 10.111111 | 9.555556 |
| 2 | 9.555556 | 9.523256 | 9.539406 |
| 3 | 9.539406 | 9.539378 | 9.539392 |
| 4 | 9.539392 | 9.539392 | 9.539392 |
After 4 passes the guess is 9.539392, already matching √91 = 9.539392 to six decimal places.
Exact form from the prime factors
91 = 7 × 13. No prime appears twice, so nothing can be pulled out of the root -- √91 is already in simplest form.
Frequently Asked Questions
What is the square root of 91?
The square root of 91 is approximately 9.539392, or exactly √91 in simplified radical form.
Is 91 a perfect square?
No, 91 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √91 between?
Between 9 and 10, because 9² = 81 and 10² = 100.
What are the square roots of the numbers right next to 91?
√90 is about 9.4868, and √92 is about 9.5917. 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 √91 squared?
Exactly 91 -- squaring undoes the square root. Squaring the rounded decimal 9.539392 gives 91, which is why the radical form √91 is the exact answer and the decimal is an approximation.
What is the negative square root of 91?
−9.539392. Every positive number has two square roots, one positive and one negative, because (−9.539392)² is also 91. The radical sign √ by convention means the positive one.
More Number Theory
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