Question

Show that $5+3\sqrt{5}$ is irrational.

Answer 1

Let us assume $5+3\sqrt{5}$ is rational

$5+3\sqrt{5}=\frac{a}{b}$

$3\sqrt{5}=\frac{a}{b}-5$

$\sqrt{5}=\frac{a}{3b}-\frac{5}{3}$

here $\sqrt{5}$ is lrrational

$\frac{a}{3b}-\frac{5}{5}$ is rational

i,e which condradicts the statement

$\therefore$ $5+3\sqrt{5}$ is irrational

Hence proved