# How do you simplify #sqrt41#?

##### 1 Answer

Jun 11, 2017

#### Answer:

#### Explanation:

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

#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

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

We can get rational approximations for

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#