Question

prove that 5√2/3 is irrational​

Answer 1

$\displaystyle \textsf{First we have to assume that \ \frac{5\sqrt{2}}{3} \ is a rational number.}\\ \\ \\ \textsf{Let \ x=\frac{5\sqrt{2}}{3} , where x is rational as we assumed.}$

$\displaystyle x=\frac{5\sqrt{2}}{3}\\ \\ \\ x=\frac{5}{3}\sqrt{2}\\ \\ \\ x \times \frac{3}{5}=\sqrt{2}\\ \\ \\ \frac{3x}{5}=\sqrt{2}$

$\textsf{Seems that \sqrt{2} can be written as a fraction at the last step.} \\ \\ \textsf{But it is irrational, then how can we write it as a fraction?!}\\ \\ \\ \textsf{Thus a contradiction occurs here.}\\ \\ \\ \\ \textsf{Hence proved...!!!}$