Question

Show that $\sqrt{5}+3\sqrt{5}$ 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 irrational & $\frac{a}{3b}-\frac{5}{3}$ is rational.

i.e., which contradicts the statement

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

Hence proved.