Question

Simplify fully 2√5 / 2+√5 ?

Answer 1

`10-4sqrt5`

Explanation:

`(2sqrt5)/(2+sqrt5)`

`"to simplify we require the denominator to be rational"`

`"to achieve this multiply the numerator/denominator by the "color(blue)"conjugate"" of "2+sqrt5`

`"the conjugate of " 2+sqrt5" is "2color(red)(-)sqrt5`

`"note that "`

`(2+sqrt5)(2-sqrt5)larrcolor(blue)"expand using FOIL"`

`=4cancel(+2sqrt5)cancel(-2sqrt5)-5=-1larrcolor(blue)"rational"`

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

`=(4sqrt5-10)/(-1)=10-4sqrt5`