Question

Simplify each of the following expressions: (i) `(3+sqrt(3))(2+sqrt(2))` (ii) `(3+sqrt(3))(3-sqrt(3))` (iii) `(sqrt(5)+sqrt(2))^2`(iv) `(sqrt(5)-sqrt(2))(sqrt(5)+sqrt(2))`

Answer 1

(i) `(3+ sqrt3)(2+ sqrt2)`
`3(2+ sqrt2) + sqrt3(2+ sqrt2)`
`= 6 + 3sqrt2 + 2sqrt3 + sqrt6`
(ii) `(3+ sqrt3)(3- sqrt3)`
`=3(3-sqrt 3) + sqrt3(3 - sqrt3)`
`= 9 cancel( - 3sqrt3) + cancel(3sqrt3) - (sqrt 3)^2`
`= 9 - 3 = 6`
(iii) `(sqrt 5 + sqrt 2 )^2`
`= (sqrt 5)^2 + (sqrt 2)^2 + 2(sqrt5)(sqrt2)`
`=5+2+2sqrt10`
`= 7 + 2sqrt10`
(iv) `(sqrt5 - sqrt2)(sqrt5 + sqrt2)`
`= (sqrt 5)^2 - (sqrt2)^2`
using identity `(a+b)(a-b) = a^2 - b^2`
`=5-2 =3`