Question

Prove that 2*√3/5 is an irrational number.​

Answer 1

Step-by-step explanation:

2*√3/5

=2√3/5

let √3 be rational in the form of p/q

√3=p/q , here p&q are co-prime integers & q is not equal to zero.

squaring both sides

3=p²/q²

p²=3q²

therefore 3 is a factor of p² as well as p - - -1

as 3 is a factor of p

p=3r

p²=9r²

but

p²=3q²

so,

3q²=9r²

therefore 3 is a factor of q² as well as q - - -2

from 1&2

3 is a factor of both p&q

so p&q are not co-prime

therefore p/q is not rational

so, √3 is irrational.

since √3 is irrational

therefore

2√3/5 is irrational

(rational *irrational)/rational= irrational