Square Root of 65
The square root of 65 is approximately 8.062258, or exactly √65 in simplified radical form. No, 65 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √65 sits
8² = 64 and 9² = 81, and 65 falls between them, so √65 is between 8 and 9. It's closer to 8 (1 above 64, 16 below 81), which matches the decimal 8.062258.
| k | k² | Compared to 65 |
|---|---|---|
| 6 | 36 | 29 below |
| 7 | 49 | 16 below |
| 8 | 64 | 1 below |
| 9 | 81 | 16 above |
| 10 | 100 | 35 above |
| 11 | 121 | 56 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 8), divide 65 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 65 ÷ x | New guess (x + 65÷x) ÷ 2 |
|---|---|---|---|
| 1 | 8 | 8.125 | 8.0625 |
| 2 | 8.0625 | 8.062016 | 8.062258 |
| 3 | 8.062258 | 8.062258 | 8.062258 |
| 4 | 8.062258 | 8.062258 | 8.062258 |
After 4 passes the guess is 8.062258, already matching √65 = 8.062258 to six decimal places.
Exact form from the prime factors
65 = 5 × 13. No prime appears twice, so nothing can be pulled out of the root -- √65 is already in simplest form.
Frequently Asked Questions
What is the square root of 65?
The square root of 65 is approximately 8.062258, or exactly √65 in simplified radical form.
Is 65 a perfect square?
No, 65 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √65 between?
Between 8 and 9, because 8² = 64 and 9² = 81.
What are the square roots of the numbers right next to 65?
√64 is exactly 8, and √66 is about 8.124. 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 √65 squared?
Exactly 65 -- squaring undoes the square root. Squaring the rounded decimal 8.062258 gives 65, which is why the radical form √65 is the exact answer and the decimal is an approximation.
What is the negative square root of 65?
−8.062258. Every positive number has two square roots, one positive and one negative, because (−8.062258)² is also 65. The radical sign √ by convention means the positive one.
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