Question

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

Answer 1

See below.

Explanation:

`4sqrt(5)` is also expressed as `4` times `sqrt(5)`, or `4*sqrt(5)`.

If you were to type `sqrt(5)` in your calculator, you would get a value of `2.2360679775`. Then, take this and multiply it by `4` to get the final answer, `8.94427191`.

I hope that helps!

Answer 2

`8.9442`

Explanation:

`4sqrt5=4*2.2361=8.9442`

Answer 3

`4sqrt(5) ~~ 51841/5796 ~~ 8.94427191`

Explanation:

Note that `(4sqrt(5))^2 = 4^2*5 = 16*5 = 80 < 81 = 9^2`

So `4sqrt(5)` is an irrational number a little less than `9`.

Note that in general:

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

So we can put `a=9` and `b=-1` to find:

`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 `4sqrt(5)`.

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`