Simplify each of the following by rationalising the denominator; `1/(5+sqrt(2))` (ii) `(5+sqrt(6))/(5-sqrt(6))` (iii) `(7+3sqrt(5))/(7-3sqrt(5))` (iv) `(2sqrt(3)-sqrt(5))/(2sqrt(2)+3sqrt(3))`

Explanation

1)`1/(5+sqrt2)=(5-sqrt2)/(5^2-sqrt2^2)=(5-sqrt2)/(5-4)=5-sqrt2`

2)`(5+sqrt6)/(5-sqrt6)=(5+sqrt6)^2/(5^2-sqrt6^2)`

`=(25+6+10sqrt6)/(25-6)=(31+10sqrt6)/19`

3)`(7+3sqrt5)/(7-3sqrt5)=(7+3sqrt5)^2/(7^2-(3sqrt5)^2)`

`=(49+45+14sqrt5)/(49-45)=(94+14sqrt5)/4`.