Square Root of 96
The square root of 96 is approximately 9.797959, or exactly 4√6 in simplified radical form. No, 96 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √96 sits
9² = 81 and 10² = 100, and 96 falls between them, so √96 is between 9 and 10. It's closer to 10 (15 above 81, 4 below 100), which matches the decimal 9.797959.
| k | k² | Compared to 96 |
|---|---|---|
| 7 | 49 | 47 below |
| 8 | 64 | 32 below |
| 9 | 81 | 15 below |
| 10 | 100 | 4 above |
| 11 | 121 | 25 above |
| 12 | 144 | 48 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 9), divide 96 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 96 ÷ x | New guess (x + 96÷x) ÷ 2 |
|---|---|---|---|
| 1 | 9 | 10.666667 | 9.833333 |
| 2 | 9.833333 | 9.762712 | 9.798023 |
| 3 | 9.798023 | 9.797895 | 9.797959 |
| 4 | 9.797959 | 9.797959 | 9.797959 |
After 4 passes the guess is 9.797959, already matching √96 = 9.797959 to six decimal places.
Exact form from the prime factors
96 = 2^5 × 3. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √96 = 4√6. That's the exact value; the decimal 9.797959 is a rounding of it.
Frequently Asked Questions
What is the square root of 96?
The square root of 96 is approximately 9.797959, or exactly 4√6 in simplified radical form.
Is 96 a perfect square?
No, 96 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √96 between?
Between 9 and 10, because 9² = 81 and 10² = 100.
What are the square roots of the numbers right next to 96?
√95 is about 9.7468, and √97 is about 9.8489. 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 √96 squared?
Exactly 96 -- squaring undoes the square root. Squaring the rounded decimal 9.797959 gives 96, which is why the radical form 4√6 is the exact answer and the decimal is an approximation.
What is the negative square root of 96?
−9.797959. Every positive number has two square roots, one positive and one negative, because (−9.797959)² is also 96. The radical sign √ by convention means the positive one.
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