Question

Rationales the denominator and simplify:

`(4sqrt3 + 5sqrt2)/(sqrt48 + sqrt18)`

Answer 1

We know that rationalization factor for `sqrt48 + sqrt18` is `sqrt48 - sqrt18`. We will multiply numerator and denominator of the given expression `(4sqrt3 + 5sqrt2)/(sqrt48 + sqrt18)` by `sqrt48 - sqrt18` to get

`(4sqrt3 + 5sqrt2)/(sqrt48 + sqrt18) xx (sqrt48 - sqrt18)/(sqrt48 - sqrt18) = (4 xx sqrt3 xx sqrt48 - 4 sqrt3 xx sqrt18 + 5 xx sqrt2 xx sqrt48 - 5 xx sqrt2 xx sqrt18)/((sqrt48)^2 - (sqrt18)^2)`

` = (4sqrt(3 xx 48) - 4 xx sqrt(3 xx 18) + 5 xx sqrt(2 xx 48) - 5 xx sqrt(2 xx 18))/(48  - 18)`

`= (4sqrt144 - 4sqrt54 + 5sqrt(96) - 5sqrt36)/30`

`= (4 xx 12 - 4 xx sqrt9 xx sqrt6 + 5 xx sqrt16 xx sqrt6 - 5sqrt36)/30`

`= (48 - 4 xx 3 xx sqrt6 + 5 xx 4 xx sqrt6 - 5 xx 6)/30`

`= (48 - 12sqrt6 + 20sqrt6 - 30)/30`

`= (18 + 8sqrt6)/30`

`= (9 + 4sqrt6)/15`

Hence the given expression is simplified to `(9 + 4sqrt6)/15`