Square Root of 8
The square root of 8 is approximately 2.828427, or exactly 2√2 in simplified radical form. No, 8 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √8 sits
2² = 4 and 3² = 9, and 8 falls between them, so √8 is between 2 and 3. It's closer to 3 (4 above 4, 1 below 9), which matches the decimal 2.828427.
| k | k² | Compared to 8 |
|---|---|---|
| 1 | 1 | 7 below |
| 2 | 4 | 4 below |
| 3 | 9 | 1 above |
| 4 | 16 | 8 above |
| 5 | 25 | 17 above |
| 6 | 36 | 28 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 2), divide 8 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 8 ÷ x | New guess (x + 8÷x) ÷ 2 |
|---|---|---|---|
| 1 | 2 | 4 | 3 |
| 2 | 3 | 2.666667 | 2.833333 |
| 3 | 2.833333 | 2.823529 | 2.828431 |
| 4 | 2.828431 | 2.828423 | 2.828427 |
After 4 passes the guess is 2.828427, already matching √8 = 2.828427 to six decimal places.
Exact form from the prime factors
8 = 2^3. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √8 = 2√2. That's the exact value; the decimal 2.828427 is a rounding of it.
Frequently Asked Questions
What is the square root of 8?
The square root of 8 is approximately 2.828427, or exactly 2√2 in simplified radical form.
Is 8 a perfect square?
No, 8 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √8 between?
Between 2 and 3, because 2² = 4 and 3² = 9.
What are the square roots of the numbers right next to 8?
√7 is about 2.6458, and √9 is exactly 3. 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 √8 squared?
Exactly 8 -- squaring undoes the square root. Squaring the rounded decimal 2.828427 gives 8, which is why the radical form 2√2 is the exact answer and the decimal is an approximation.
What is the negative square root of 8?
−2.828427. Every positive number has two square roots, one positive and one negative, because (−2.828427)² is also 8. The radical sign √ by convention means the positive one.
More Number Theory
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