# How do you calculate #4sqrt(5)# ?

##### 3 Answers

See below.

#### Explanation:

If you were to type

I hope that helps!

#### Explanation:

#### Explanation:

Note that

So

Note that in general:

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

So we can put

#4sqrt(5) = sqrt(80) = sqrt(9^2-1) = 9-1/(18-1/(18-1/(18-1/(18-...))))#

We can terminate this generalised continued fraction early to get rational approximations to

For example:

#4sqrt(5) ~~ 9-1/18 = 8.9bar(4)#

#4sqrt(5) ~~ 9-1/(18-1/18) = 9-18/323 = 2889/323 ~~ 8.944272#

#4sqrt(5) ~~ 9-1/(18-1/(18-1/18)) = 9-1/(18-18/323) = 9-323/5796 = 51841/5796 ~~ 8.94427191#