Square Root of 68
The square root of 68 is approximately 8.246211, or exactly 2√17 in simplified radical form. No, 68 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √68 sits
8² = 64 and 9² = 81, and 68 falls between them, so √68 is between 8 and 9. It's closer to 8 (4 above 64, 13 below 81), which matches the decimal 8.246211.
| k | k² | Compared to 68 |
|---|---|---|
| 6 | 36 | 32 below |
| 7 | 49 | 19 below |
| 8 | 64 | 4 below |
| 9 | 81 | 13 above |
| 10 | 100 | 32 above |
| 11 | 121 | 53 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 8), divide 68 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 68 ÷ x | New guess (x + 68÷x) ÷ 2 |
|---|---|---|---|
| 1 | 8 | 8.5 | 8.25 |
| 2 | 8.25 | 8.242424 | 8.246212 |
| 3 | 8.246212 | 8.24621 | 8.246211 |
| 4 | 8.246211 | 8.246211 | 8.246211 |
After 4 passes the guess is 8.246211, already matching √68 = 8.246211 to six decimal places.
Exact form from the prime factors
68 = 2^2 × 17. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √68 = 2√17. That's the exact value; the decimal 8.246211 is a rounding of it.
Frequently Asked Questions
What is the square root of 68?
The square root of 68 is approximately 8.246211, or exactly 2√17 in simplified radical form.
Is 68 a perfect square?
No, 68 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √68 between?
Between 8 and 9, because 8² = 64 and 9² = 81.
What are the square roots of the numbers right next to 68?
√67 is about 8.1854, and √69 is about 8.3066. 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 √68 squared?
Exactly 68 -- squaring undoes the square root. Squaring the rounded decimal 8.246211 gives 68, which is why the radical form 2√17 is the exact answer and the decimal is an approximation.
What is the negative square root of 68?
−8.246211. Every positive number has two square roots, one positive and one negative, because (−8.246211)² is also 68. The radical sign √ by convention means the positive one.
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