Square Root of 88
The square root of 88 is approximately 9.380832, or exactly 2√22 in simplified radical form. No, 88 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √88 sits
9² = 81 and 10² = 100, and 88 falls between them, so √88 is between 9 and 10. It's closer to 9 (7 above 81, 12 below 100), which matches the decimal 9.380832.
| k | k² | Compared to 88 |
|---|---|---|
| 7 | 49 | 39 below |
| 8 | 64 | 24 below |
| 9 | 81 | 7 below |
| 10 | 100 | 12 above |
| 11 | 121 | 33 above |
| 12 | 144 | 56 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 9), divide 88 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 88 ÷ x | New guess (x + 88÷x) ÷ 2 |
|---|---|---|---|
| 1 | 9 | 9.777778 | 9.388889 |
| 2 | 9.388889 | 9.372781 | 9.380835 |
| 3 | 9.380835 | 9.380828 | 9.380832 |
| 4 | 9.380832 | 9.380832 | 9.380832 |
After 4 passes the guess is 9.380832, already matching √88 = 9.380832 to six decimal places.
Exact form from the prime factors
88 = 2^3 × 11. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √88 = 2√22. That's the exact value; the decimal 9.380832 is a rounding of it.
Frequently Asked Questions
What is the square root of 88?
The square root of 88 is approximately 9.380832, or exactly 2√22 in simplified radical form.
Is 88 a perfect square?
No, 88 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √88 between?
Between 9 and 10, because 9² = 81 and 10² = 100.
What are the square roots of the numbers right next to 88?
√87 is about 9.3274, and √89 is about 9.434. 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 √88 squared?
Exactly 88 -- squaring undoes the square root. Squaring the rounded decimal 9.380832 gives 88, which is why the radical form 2√22 is the exact answer and the decimal is an approximation.
What is the negative square root of 88?
−9.380832. Every positive number has two square roots, one positive and one negative, because (−9.380832)² is also 88. The radical sign √ by convention means the positive one.
More Number Theory
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