Square Root of 47
The square root of 47 is approximately 6.855655, or exactly √47 in simplified radical form. No, 47 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √47 sits
6² = 36 and 7² = 49, and 47 falls between them, so √47 is between 6 and 7. It's closer to 7 (11 above 36, 2 below 49), which matches the decimal 6.855655.
| k | k² | Compared to 47 |
|---|---|---|
| 4 | 16 | 31 below |
| 5 | 25 | 22 below |
| 6 | 36 | 11 below |
| 7 | 49 | 2 above |
| 8 | 64 | 17 above |
| 9 | 81 | 34 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 6), divide 47 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 47 ÷ x | New guess (x + 47÷x) ÷ 2 |
|---|---|---|---|
| 1 | 6 | 7.833333 | 6.916667 |
| 2 | 6.916667 | 6.795181 | 6.855924 |
| 3 | 6.855924 | 6.855386 | 6.855655 |
| 4 | 6.855655 | 6.855655 | 6.855655 |
After 4 passes the guess is 6.855655, already matching √47 = 6.855655 to six decimal places.
Exact form from the prime factors
47 = 47. No prime appears twice, so nothing can be pulled out of the root -- √47 is already in simplest form.
Frequently Asked Questions
What is the square root of 47?
The square root of 47 is approximately 6.855655, or exactly √47 in simplified radical form.
Is 47 a perfect square?
No, 47 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √47 between?
Between 6 and 7, because 6² = 36 and 7² = 49.
What are the square roots of the numbers right next to 47?
√46 is about 6.7823, and √48 is about 6.9282. 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 √47 squared?
Exactly 47 -- squaring undoes the square root. Squaring the rounded decimal 6.855655 gives 47, which is why the radical form √47 is the exact answer and the decimal is an approximation.
What is the negative square root of 47?
−6.855655. Every positive number has two square roots, one positive and one negative, because (−6.855655)² is also 47. The radical sign √ by convention means the positive one.
More Number Theory
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