Square Root of 99
The square root of 99 is approximately 9.949874, or exactly 3√11 in simplified radical form. No, 99 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √99 sits
9² = 81 and 10² = 100, and 99 falls between them, so √99 is between 9 and 10. It's closer to 10 (18 above 81, 1 below 100), which matches the decimal 9.949874.
| k | k² | Compared to 99 |
|---|---|---|
| 7 | 49 | 50 below |
| 8 | 64 | 35 below |
| 9 | 81 | 18 below |
| 10 | 100 | 1 above |
| 11 | 121 | 22 above |
| 12 | 144 | 45 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 9), divide 99 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 99 ÷ x | New guess (x + 99÷x) ÷ 2 |
|---|---|---|---|
| 1 | 9 | 11 | 10 |
| 2 | 10 | 9.9 | 9.95 |
| 3 | 9.95 | 9.949749 | 9.949874 |
| 4 | 9.949874 | 9.949874 | 9.949874 |
After 4 passes the guess is 9.949874, already matching √99 = 9.949874 to six decimal places.
Exact form from the prime factors
99 = 3^2 × 11. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √99 = 3√11. That's the exact value; the decimal 9.949874 is a rounding of it.
Frequently Asked Questions
What is the square root of 99?
The square root of 99 is approximately 9.949874, or exactly 3√11 in simplified radical form.
Is 99 a perfect square?
No, 99 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √99 between?
Between 9 and 10, because 9² = 81 and 10² = 100.
What are the square roots of the numbers right next to 99?
√98 is about 9.8995, and √100 is exactly 10. 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 √99 squared?
Exactly 99 -- squaring undoes the square root. Squaring the rounded decimal 9.949874 gives 99, which is why the radical form 3√11 is the exact answer and the decimal is an approximation.
What is the negative square root of 99?
−9.949874. Every positive number has two square roots, one positive and one negative, because (−9.949874)² is also 99. The radical sign √ by convention means the positive one.
More Number Theory
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