√41 at a glance
- Exact value
- √41
- Decimal (10 places)
- 6.4031242374
- Rounded
- 6.4 · 6.40 · 6.403
- Perfect square?
- No — between 6² and 7²
- Rational?
- Irrational
- Both square roots
- ±6.403124
- Prime factorization
- 41
- Cube root
- 3.448217
How to simplify √41
41 is a prime number, so its only factors are 1 and 41. There is no perfect-square factor to pull out, which means √41 is already in its simplest radical form.
The square root of any prime is irrational. If √41 were a fraction a/b in lowest terms, then a² = 41b², so 41 would divide a — and then 41 would divide b too, contradicting “lowest terms.” That is why the decimal 6.4031242374 is only a rounded value.
Where √41 sits between perfect squares
36 = 6² and 49 = 7² are the nearest perfect squares, so √41 lies between 6 and 7. 41 is 5 above 36 and 8 below 49, so the root is closer to 6.
- Straight line between 36 and 49: 6.3846 (0.29% low)
- Tangent from 6, i.e. 6 + 5 ÷ 12: 6.4167 (0.21% high)
- Tangent from 7, i.e. 7 − 8 ÷ 14: 6.4286 (0.4% high)
For √41 the tangent at 6 wins, missing by only 0.0135. Tangent estimates shine when the number sits close to a perfect square — here 41 is just 5 above 36.
Finding √41 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 41: following the tangent line down to zero simplifies to averaging x with 41 ÷ x.
Start from the nearest whole number, 6 (6² = 36):
| Step | Guess x | 41 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 6.0000000000 | 6.8333333333 | 6.4166666667 | 1 |
| 2 | 6.4166666667 | 6.3896103896 | 6.4031385281 | 4 |
| 3 | 6.4031385281 | 6.4031099468 | 6.4031242374 | all 10 shown |
The count of correct decimals went 1, 4 and all 10 over 3 steps — roughly doubling each time — until the guess matched √41 = 6.4031242374 to every decimal shown.
√41 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √41 the pattern is [6; 2, 2, 12] with the block of 3 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √41 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 6/1 | 6.0000000000 | 4.0 × 10⁻¹ |
| 13/2 | 6.5000000000 | 9.7 × 10⁻² |
| 32/5 | 6.4000000000 | 3.1 × 10⁻³ |
| 397/62 | 6.4032258065 | 1.0 × 10⁻⁴ |
| 826/129 | 6.4031007752 | 2.3 × 10⁻⁵ |
| 2,049/320 | 6.4031250000 | 7.6 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 41y² = 1. Its smallest solution in positive whole numbers is x = 2,049, y = 320. Because the period is odd, the equation with −1 on the right also has a solution: 32² − 41 × 5² = −1.
√41 in geometry and everyday measurements
- A square room or garden bed covering 41 square feet measures about 6.4 ft (6 ft 5 in) along each wall.
- 41 = 4² + 5², so by the Pythagorean theorem √41 is the diagonal of a 4 × 5 rectangle — and the distance between the points (0, 0) and (4, 5) on a grid.
Square roots near √41 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √38 | √38 | 6.1644 | No |
| √39 | √39 | 6.2450 | No |
| √40 | 2√10 | 6.3246 | No |
| √41 | √41 | 6.4031 | No |
| √42 | √42 | 6.4807 | No |
| √43 | √43 | 6.5574 | No |
| √44 | 2√11 | 6.6332 | No |
- The cube root of 41 is about 3.448217.
- Four times the radicand doubles the root: √164 = 2 × √41 ≈ 12.806248.
Frequently asked questions
What is the square root of 41?
The square root of 41 is √41, about 6.4031242374. The negative root, −6.403124, also squares to 41.
Is the square root of 41 rational or irrational?
Irrational. 41 is not a perfect square — it falls between 36 and 49 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √41 be simplified?
No. 41 is prime, so there is no perfect square to take out of the radical.
What is √41 rounded to two decimal places?
√41 ≈ 6.40 to two decimal places (6.4 to one, 6.403 to three). Check: 6.40² = 40.96, close to 41.