Question

prove that the difference and quotient of (3+2root 3) and (3-2root 3) are irrational.

Answer 1

The difference of (3+2√3) and(3-2√3) is: 3+2√3-3+2√3=4√3 which is irrational.
The quotient of (3+2√3) and(3-2√30 is:
(3+2√3)/(3-2√3)
=(3+2√3)(3+2√3)/(3-2√3)(3+2√3)
=(3+2√3)²/[(3)²-(2√3)²]
={9+2.3.2√3+(2√3)²}/(9-12)
=(9+12√3+12)/(-3)
=(21+12√3)/(-3)
=3(7+4√3)/(-3)
=7+4√3
which is irrational.