Question

How do you simplify `(sqrt(3) - 2) / (-5*sqrt(3)+8)`?

Answer 1

`color(green)((1+2sqrt3)/11`

Explanation:

`(sqrt3-2)/(-5*sqrt3+8)`

`(sqrt3-2)/(-5*sqrt3+8) xx (-5sqrt3-8)/(-5sqrt3-8)` (conjugate)

`color(white)(aaaaaaaaaaaaa)` `sqrt3-2`
`color(white)(aaaaaaaa)` `xx underline(-5sqrt3-8)`
`color(white)(aaaaaaaaaaaaa)` `(-5*3)+10sqrt3`
`color(white)(aaaaaaaaaaaaaaaaaaaaaa)` `-8sqrt3+16`
`color(white)(aaaaaaaaaaaaaaaaaa)` `overline(-15+2sqrt3+16)`

`color(white)(aaaaaaaaaaaaa)` `color(green)(1+2sqrt3` numerator

`color(white)(aaaaaaaaaaaaa)` `-5sqrt3+8`
`color(white)(aaaaaaaaaaa)` `xx underline(-5sqrt3-8)`
`color(white)(aaaaaaaaaaaaa)` `(25*3)-40sqrt3`
`color(white)(aaaaaaaaaaaaaaaaaaaa)` `ul(+40sqrt3-64)`
`color(white)(aaaaaaaaaaaaaaaaaa)` `75-64`

`color(white)(aaaaaaaaaaaaa)` `color(green)(11` denominator

`:.color(green)(=(1+2sqrt3)/11`

Check:

`color(red)((sqrt3-2)/(-5*sqrt3+8)=0.405827418`

`color(red)((1+2sqrt3)/11=0.405827418`