Square Root of 56
The square root of 56 is approximately 7.483315, or exactly 2√14 in simplified radical form. No, 56 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √56 sits
7² = 49 and 8² = 64, and 56 falls between them, so √56 is between 7 and 8. It's closer to 7 (7 above 49, 8 below 64), which matches the decimal 7.483315.
| k | k² | Compared to 56 |
|---|---|---|
| 5 | 25 | 31 below |
| 6 | 36 | 20 below |
| 7 | 49 | 7 below |
| 8 | 64 | 8 above |
| 9 | 81 | 25 above |
| 10 | 100 | 44 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 7), divide 56 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 56 ÷ x | New guess (x + 56÷x) ÷ 2 |
|---|---|---|---|
| 1 | 7 | 8 | 7.5 |
| 2 | 7.5 | 7.466667 | 7.483333 |
| 3 | 7.483333 | 7.483296 | 7.483315 |
| 4 | 7.483315 | 7.483315 | 7.483315 |
After 4 passes the guess is 7.483315, already matching √56 = 7.483315 to six decimal places.
Exact form from the prime factors
56 = 2^3 × 7. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √56 = 2√14. That's the exact value; the decimal 7.483315 is a rounding of it.
Frequently Asked Questions
What is the square root of 56?
The square root of 56 is approximately 7.483315, or exactly 2√14 in simplified radical form.
Is 56 a perfect square?
No, 56 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √56 between?
Between 7 and 8, because 7² = 49 and 8² = 64.
What are the square roots of the numbers right next to 56?
√55 is about 7.4162, and √57 is about 7.5498. 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 √56 squared?
Exactly 56 -- squaring undoes the square root. Squaring the rounded decimal 7.483315 gives 56, which is why the radical form 2√14 is the exact answer and the decimal is an approximation.
What is the negative square root of 56?
−7.483315. Every positive number has two square roots, one positive and one negative, because (−7.483315)² is also 56. The radical sign √ by convention means the positive one.
More Number Theory
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