Question

☺️☺️50 points_______

♣️ prove that the difference and quotient of (32 root 3) and (3-2root3) are irrational.​

Answer 1

Hey mate,..

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

Hope it will help you

Answer 2

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√3) 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 is an irrational number

= - (7+4√3) is an irrational number

The quotient of (3+2√3) and(3-2√3) is an irrational number.