Square Root of 41
The square root of 41 is approximately 6.403124, or exactly √41 in simplified radical form. No, 41 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √41 sits
6² = 36 and 7² = 49, and 41 falls between them, so √41 is between 6 and 7. It's closer to 6 (5 above 36, 8 below 49), which matches the decimal 6.403124.
| k | k² | Compared to 41 |
|---|---|---|
| 4 | 16 | 25 below |
| 5 | 25 | 16 below |
| 6 | 36 | 5 below |
| 7 | 49 | 8 above |
| 8 | 64 | 23 above |
| 9 | 81 | 40 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 6), divide 41 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 41 ÷ x | New guess (x + 41÷x) ÷ 2 |
|---|---|---|---|
| 1 | 6 | 6.833333 | 6.416667 |
| 2 | 6.416667 | 6.38961 | 6.403139 |
| 3 | 6.403139 | 6.40311 | 6.403124 |
| 4 | 6.403124 | 6.403124 | 6.403124 |
After 4 passes the guess is 6.403124, already matching √41 = 6.403124 to six decimal places.
Exact form from the prime factors
41 = 41. No prime appears twice, so nothing can be pulled out of the root -- √41 is already in simplest form.
Frequently Asked Questions
What is the square root of 41?
The square root of 41 is approximately 6.403124, or exactly √41 in simplified radical form.
Is 41 a perfect square?
No, 41 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √41 between?
Between 6 and 7, because 6² = 36 and 7² = 49.
What are the square roots of the numbers right next to 41?
√40 is about 6.3246, and √42 is about 6.4807. 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 √41 squared?
Exactly 41 -- squaring undoes the square root. Squaring the rounded decimal 6.403124 gives 41, which is why the radical form √41 is the exact answer and the decimal is an approximation.
What is the negative square root of 41?
−6.403124. Every positive number has two square roots, one positive and one negative, because (−6.403124)² is also 41. The radical sign √ by convention means the positive one.
More Number Theory
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