Square Root of 97
The square root of 97 is approximately 9.848858, or exactly √97 in simplified radical form. No, 97 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √97 sits
9² = 81 and 10² = 100, and 97 falls between them, so √97 is between 9 and 10. It's closer to 10 (16 above 81, 3 below 100), which matches the decimal 9.848858.
| k | k² | Compared to 97 |
|---|---|---|
| 7 | 49 | 48 below |
| 8 | 64 | 33 below |
| 9 | 81 | 16 below |
| 10 | 100 | 3 above |
| 11 | 121 | 24 above |
| 12 | 144 | 47 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 9), divide 97 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 97 ÷ x | New guess (x + 97÷x) ÷ 2 |
|---|---|---|---|
| 1 | 9 | 10.777778 | 9.888889 |
| 2 | 9.888889 | 9.808989 | 9.848939 |
| 3 | 9.848939 | 9.848777 | 9.848858 |
| 4 | 9.848858 | 9.848858 | 9.848858 |
After 4 passes the guess is 9.848858, already matching √97 = 9.848858 to six decimal places.
Exact form from the prime factors
97 = 97. No prime appears twice, so nothing can be pulled out of the root -- √97 is already in simplest form.
Frequently Asked Questions
What is the square root of 97?
The square root of 97 is approximately 9.848858, or exactly √97 in simplified radical form.
Is 97 a perfect square?
No, 97 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √97 between?
Between 9 and 10, because 9² = 81 and 10² = 100.
What are the square roots of the numbers right next to 97?
√96 is about 9.798, and √98 is about 9.8995. 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 √97 squared?
Exactly 97 -- squaring undoes the square root. Squaring the rounded decimal 9.848858 gives 97, which is why the radical form √97 is the exact answer and the decimal is an approximation.
What is the negative square root of 97?
−9.848858. Every positive number has two square roots, one positive and one negative, because (−9.848858)² is also 97. The radical sign √ by convention means the positive one.
More Number Theory
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