Square Root of 98
The square root of 98 is approximately 9.899495, or exactly 7√2 in simplified radical form. No, 98 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √98 sits
9² = 81 and 10² = 100, and 98 falls between them, so √98 is between 9 and 10. It's closer to 10 (17 above 81, 2 below 100), which matches the decimal 9.899495.
| k | k² | Compared to 98 |
|---|---|---|
| 7 | 49 | 49 below |
| 8 | 64 | 34 below |
| 9 | 81 | 17 below |
| 10 | 100 | 2 above |
| 11 | 121 | 23 above |
| 12 | 144 | 46 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 9), divide 98 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 98 ÷ x | New guess (x + 98÷x) ÷ 2 |
|---|---|---|---|
| 1 | 9 | 10.888889 | 9.944444 |
| 2 | 9.944444 | 9.854749 | 9.899597 |
| 3 | 9.899597 | 9.899393 | 9.899495 |
| 4 | 9.899495 | 9.899495 | 9.899495 |
After 4 passes the guess is 9.899495, already matching √98 = 9.899495 to six decimal places.
Exact form from the prime factors
98 = 2 × 7^2. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √98 = 7√2. That's the exact value; the decimal 9.899495 is a rounding of it.
Frequently Asked Questions
What is the square root of 98?
The square root of 98 is approximately 9.899495, or exactly 7√2 in simplified radical form.
Is 98 a perfect square?
No, 98 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √98 between?
Between 9 and 10, because 9² = 81 and 10² = 100.
What are the square roots of the numbers right next to 98?
√97 is about 9.8489, and √99 is about 9.9499. 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 √98 squared?
Exactly 98 -- squaring undoes the square root. Squaring the rounded decimal 9.899495 gives 98, which is why the radical form 7√2 is the exact answer and the decimal is an approximation.
What is the negative square root of 98?
−9.899495. Every positive number has two square roots, one positive and one negative, because (−9.899495)² is also 98. The radical sign √ by convention means the positive one.
More Number Theory
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