Question

How do you evaluate `\frac { 1} { - \sqrt { 48} }`?

Answer 1

`1/(-sqrt(48)) = -sqrt(3)/12 ~~ -195/1351 ~~ -0.1443375`

Explanation:

Note that:

`48 = 4^2 * 3`

`sqrt(ab) = sqrt(a)sqrt(b)" "` if `a, b >= 0`

So we find:

`1/(-sqrt(48)) = -1/(sqrt(4^2 * 3))`

`color(white)(1/(-sqrt(48))) = -1/(sqrt(4^2) * sqrt(3))`

`color(white)(1/(-sqrt(48))) = -1/(4sqrt(3))`

`color(white)(1/(-sqrt(48))) = -sqrt(3)/(4sqrt(3)sqrt(3))`

`color(white)(1/(-sqrt(48))) = -sqrt(3)/12`

That is the simplest form.

This is an irrational number. To calculate rational approximations we can use:

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

with `a=7` and `b=-1`

as follows:

`sqrt(48) = sqrt(7^2-1) = 7-1/(14-1/(14-1/(14-1/(14-1/(14-...)))))`

So:

`1/(-sqrt(48)) = -1/(7-1/(14-1/(14-1/(14-1/(14-1/(14-...)))))`

`color(white)(1/(-sqrt(48))) ~~ -1/(7-1/(14-1/14)) = -1/(7-14/195) = -195/1351 ~~ -0.1443375`