Question

Prove that difference and quotient of 3+2 root3 and 3-2 root3 are irrational

Answer 1

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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.