Question

Simplify the following :

`(7sqrt(3))/(sqrt(10) + sqrt(3)) - (2sqrt(5))/(sqrt(6) + sqrt(5)) - (3sqrt(2))/(sqrt(15) + 3sqrt(2)`

Answer 1

`(7sqrt(3))/(sqrt(10) + sqrt(3)) - (2sqrt(5))/(sqrt(6) + sqrt(5)) - (3sqrt(2))/(sqrt(15) + 3sqrt(2)`

Rationalizing the denominator of each term, we have

= `(7sqrt(3)(sqrt(10) - sqrt(3)))/((sqrt(10) + sqrt(3))(sqrt(10) - sqrt(3))) - (2sqrt(5)(sqrt(6) - sqrt(5)))/((sqrt(6) + sqrt(5))(sqrt(6) - sqrt(5))) - (3sqrt(2)(3sqrt(2) - sqrt(15)))/((sqrt(15) + 3sqrt(2))(3sqrt(2)-sqrt(15))`

= `(7sqrt(3)(sqrt(10)- sqrt(3)))/(10 - 3) - (2sqrt(5)(sqrt(6) - sqrt(5)))/(6 - 5) - (3sqrt(2)(3sqrt(2) - sqrt(15)))/(18 - 15)`

= `sqrt3(sqrt10 - sqrt3) - 2sqrt5(sqrt6 - sqrt5) - sqrt2(3sqrt2 - sqrt15)`

= `sqrt30 - 3 - 2sqrt30 + 10 - 6 + sqrt30`

= 0 − 9 + 10

= 1