Question

How do you simplify `(2 -sqrt 5) /(3 +sqrt5)`?

Answer 1

`color(chocolate)(=> 11/4 - (5sqrt5)/4`

Explanation:

`(2 - sqrt5) / (3 + sqrt5)`

`=> ((2 - sqrt5) * (3 - sqrt5)) / ((3 + sqrt5) (3 - sqrt5))`

`=> (6 - 2 sqrt5 -3sqrt5 + 5 ) / (9 - 5)`

`=> (11 - 5sqrt5) / 4`

`color(chocolate)(=> 11/4 - (5sqrt5)/4`

Answer 2

`11/4-(5sqrt5)/4`

Explanation:

`"multiply the numerator/denominator by the "color(blue)"conjugate"`
`"of the denominator"`

`"the conjugate of "3+sqrt5" is "3color(red)(-)sqrt5`

`=((2-sqrt5)(3-sqrt5))/((3+sqrt5)(3-sqrt5))`

`"expand factors on numerator/denominator"`

`=(6-5sqrt5+5)/(9-5)`

`=(11-5sqrt5)/4=11/4-(5sqrt5)/4`