Square Root of 67
The square root of 67 is approximately 8.185353, or exactly √67 in simplified radical form. No, 67 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √67 sits
8² = 64 and 9² = 81, and 67 falls between them, so √67 is between 8 and 9. It's closer to 8 (3 above 64, 14 below 81), which matches the decimal 8.185353.
| k | k² | Compared to 67 |
|---|---|---|
| 6 | 36 | 31 below |
| 7 | 49 | 18 below |
| 8 | 64 | 3 below |
| 9 | 81 | 14 above |
| 10 | 100 | 33 above |
| 11 | 121 | 54 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 8), divide 67 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 67 ÷ x | New guess (x + 67÷x) ÷ 2 |
|---|---|---|---|
| 1 | 8 | 8.375 | 8.1875 |
| 2 | 8.1875 | 8.183206 | 8.185353 |
| 3 | 8.185353 | 8.185352 | 8.185353 |
| 4 | 8.185353 | 8.185353 | 8.185353 |
After 4 passes the guess is 8.185353, already matching √67 = 8.185353 to six decimal places.
Exact form from the prime factors
67 = 67. No prime appears twice, so nothing can be pulled out of the root -- √67 is already in simplest form.
Frequently Asked Questions
What is the square root of 67?
The square root of 67 is approximately 8.185353, or exactly √67 in simplified radical form.
Is 67 a perfect square?
No, 67 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √67 between?
Between 8 and 9, because 8² = 64 and 9² = 81.
What are the square roots of the numbers right next to 67?
√66 is about 8.124, and √68 is about 8.2462. 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 √67 squared?
Exactly 67 -- squaring undoes the square root. Squaring the rounded decimal 8.185353 gives 67, which is why the radical form √67 is the exact answer and the decimal is an approximation.
What is the negative square root of 67?
−8.185353. Every positive number has two square roots, one positive and one negative, because (−8.185353)² is also 67. The radical sign √ by convention means the positive one.
More Number Theory
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