Question

prove that the difference and quotient of (3+2√3)and (3-2√3) are irreational

Answer 1

First we will prove the difference is irrational
we have,
(3+2√3)-(3-2√3)=3+2√3-3+2√3(since -×-=+)
=4√3
therefore it is irrational since √3 is irrational and as we know product of a rational( here ,4)and an irrational (here,√3) is always irrational.
Now the quotient,
(3+2√3)/(3-2√3)={(3+2√3)×(3+2√3)}/{(3-2√3)×(3+√2)}
(in the above equation I have multiplied( 3+√2) with the numerator and the denominator)
={(3+2√3)^2}/{3^2-(2√3)^2}
=(9+12+12√3)/(9-12)
=(21+12√3)/(-3)
=(-21-12√3)/(3)
now as we know the difference of a rational(here,-21)and an irrational (here,-12√3)
is always irrational therefore( -21-12√3) is irrational and we also know the quotient of a rational (here,3) and an irrational(here,-21-12√3) is always irrational
therefore (-21-12√3)/(3) is an irrational number
and hence (3+2√3)/(3-2√3) is an irrational number......

Answer 2

First we will prove the difference is irrational

we have,

(3+2√3)-(3-2√3)=3+2√3-3+2√3(since -×-=+)

=4√3

therefore it is irrational since √3 is irrational and as we know product of a rational( here ,4)and an irrational (here,√3) is always irrational.

Now the quotient,

(3+2√3)/(3-2√3)={(3+2√3)×(3+2√3)}/{(3-2√3)×(3+√2)}

(in the above equation I have multiplied( 3+√2) with the numerator and the denominator)

={(3+2√3)^2}/{3^2-(2√3)^2}

=(9+12+12√3)/(9-12)

=(21+12√3)/(-3)

=(-21-12√3)/(3)

now as we know the difference of a rational(here,-21)and an irrational (here,-12√3)

is always irrational therefore( -21-12√3) is irrational and we also know the quotient of a rational (here,3) and an irrational(here,-21-12√3) is always irrational

therefore (-21-12√3)/(3) is an irrational number

and hence (3+2√3)/(3-2√3) is an irrational number......