Question

prove that 2+3 root 2 is irrational number ​

Answer 1

hope it's clear......

Answer 2

Assume to reach the contradiction that 2+3√2 is a rational number.

Let x = 2+3√2, where x is a rational number which values the RHS.

$x=2+3\sqrt{2} \\ \\ \\ x-2=3\sqrt{2} \\ \\ \\ \displaystyle \frac{x-2}{3}=\sqrt{2}$

Consider the last line.

→  In the LHS,

    ⇒  x is rational.

    ⇒  As a rational number subtracted by another rational number gives rational number, x - 2 is rational.

    ⇒  As a rational number divided by another rational number gives rational number, (x - 2)/3 is rational.

Thus the LHS is rational. While the RHS √2 is irrational.

Hence this final step creates a contradiction.

Thus proved!!!