Square Root of 66
The square root of 66 is approximately 8.124038, or exactly √66 in simplified radical form. No, 66 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √66 sits
8² = 64 and 9² = 81, and 66 falls between them, so √66 is between 8 and 9. It's closer to 8 (2 above 64, 15 below 81), which matches the decimal 8.124038.
| k | k² | Compared to 66 |
|---|---|---|
| 6 | 36 | 30 below |
| 7 | 49 | 17 below |
| 8 | 64 | 2 below |
| 9 | 81 | 15 above |
| 10 | 100 | 34 above |
| 11 | 121 | 55 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 8), divide 66 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 66 ÷ x | New guess (x + 66÷x) ÷ 2 |
|---|---|---|---|
| 1 | 8 | 8.25 | 8.125 |
| 2 | 8.125 | 8.123077 | 8.124038 |
| 3 | 8.124038 | 8.124038 | 8.124038 |
| 4 | 8.124038 | 8.124038 | 8.124038 |
After 4 passes the guess is 8.124038, already matching √66 = 8.124038 to six decimal places.
Exact form from the prime factors
66 = 2 × 3 × 11. No prime appears twice, so nothing can be pulled out of the root -- √66 is already in simplest form.
Frequently Asked Questions
What is the square root of 66?
The square root of 66 is approximately 8.124038, or exactly √66 in simplified radical form.
Is 66 a perfect square?
No, 66 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √66 between?
Between 8 and 9, because 8² = 64 and 9² = 81.
What are the square roots of the numbers right next to 66?
√65 is about 8.0623, and √67 is about 8.1854. 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 √66 squared?
Exactly 66 -- squaring undoes the square root. Squaring the rounded decimal 8.124038 gives 66, which is why the radical form √66 is the exact answer and the decimal is an approximation.
What is the negative square root of 66?
−8.124038. Every positive number has two square roots, one positive and one negative, because (−8.124038)² is also 66. The radical sign √ by convention means the positive one.
More Number Theory
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