How do you simplify $sqrt41$ ?

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Answer 1 · 2017-06-11T15:17:34

$sqrt(41) ~~ 6.4031242374$ is an irrational number which cannot be simplified.

$41$ is a prime number, so has no square factors.

As a result its square root cannot be simplified. It is an irrational number.

We find:

$6^2 = 36 < 41 < 49 = 7^2$

So:

$6 < sqrt(41) < 7$

To get a good approximation for $sqrt(41)$ note that:

$64^2 = 4096$

So:

$sqrt(41) ~~ sqrt(40.96) = 6.4 = 32/5$

In general:

$sqrt(a^2+b) = a+b/(2a+b/(2a+b/(2a+...)))$

Putting $a=32/5$ and $b=1/25$ , we find:

$sqrt(41) = 32/5+(1/25)/(64/5+(1/25)/(64/5+(1/25)/(64/5+...)))$

We can get rational approximations for $sqrt(41)$ by truncating this continued fraction.

For example:

$sqrt(41) ~~ 32/5+(1/25)/(64/5) = 2049/320$

$sqrt(41) ~~ 32/5+(1/25)/(64/5+(1/25)/(64/5)) = 131168/20485 ~~ 6.4031242374$

How do you simplify sqrt41sqrt41?