Question

If x= (√3+√2) / (√3-√2) and y= (√3-√2) / ( √3+√2) determine x²+y²

Answer 1

(sqrt(3)+sqrt(2))^2=3+2+2sqrt(6)=5+2sqrt(6). Since x=1/y, x^2=1/y^2 and the question becomes (5+2sqrt(6))/(5-2sqrt(6))+(5-2sqrt(6))/(5+2sqrt(6)).

Each fraction can be rationalised:

(5+2sqrt(6))^2/(25-24)+(5-2sqrt(6))^2/(25-24)=

(5+2sqrt(6))^2+(5-2sqrt(6))^2=2*5^2+2*2^2*6=50+48=98.

(Example of rationalisation: (5+2sqrt(6))/(5-2sqrt(6))=

(5+2sqrt(6))(5+2sqrt(6))/(5-2sqrt(6))(5+2sqrt(6))=

(5+2sqrt(6))^2/(5^2-(2sqrt(6))^2)=

(5+2sqrt(6))^2/(25-24)=(5+2sqrt(6))^2.)

Therefore the answer is simply 98, because the irrational parts cancel out.