Square Root of 69
The square root of 69 is approximately 8.306624, or exactly √69 in simplified radical form. No, 69 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √69 sits
8² = 64 and 9² = 81, and 69 falls between them, so √69 is between 8 and 9. It's closer to 8 (5 above 64, 12 below 81), which matches the decimal 8.306624.
| k | k² | Compared to 69 |
|---|---|---|
| 6 | 36 | 33 below |
| 7 | 49 | 20 below |
| 8 | 64 | 5 below |
| 9 | 81 | 12 above |
| 10 | 100 | 31 above |
| 11 | 121 | 52 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 8), divide 69 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 69 ÷ x | New guess (x + 69÷x) ÷ 2 |
|---|---|---|---|
| 1 | 8 | 8.625 | 8.3125 |
| 2 | 8.3125 | 8.300752 | 8.306626 |
| 3 | 8.306626 | 8.306622 | 8.306624 |
| 4 | 8.306624 | 8.306624 | 8.306624 |
After 4 passes the guess is 8.306624, already matching √69 = 8.306624 to six decimal places.
Exact form from the prime factors
69 = 3 × 23. No prime appears twice, so nothing can be pulled out of the root -- √69 is already in simplest form.
Frequently Asked Questions
What is the square root of 69?
The square root of 69 is approximately 8.306624, or exactly √69 in simplified radical form.
Is 69 a perfect square?
No, 69 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √69 between?
Between 8 and 9, because 8² = 64 and 9² = 81.
What are the square roots of the numbers right next to 69?
√68 is about 8.2462, and √70 is about 8.3666. 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 √69 squared?
Exactly 69 -- squaring undoes the square root. Squaring the rounded decimal 8.306624 gives 69, which is why the radical form √69 is the exact answer and the decimal is an approximation.
What is the negative square root of 69?
−8.306624. Every positive number has two square roots, one positive and one negative, because (−8.306624)² is also 69. The radical sign √ by convention means the positive one.
More Number Theory
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