Question

Show that the following numbers are irrational.

$$7\sqrt{5}$$

Answer 1

 Let us assume that ${\displaystyle \frac{1}{\sqrt{2}}}$ is rational .Then , there exist positive co primes a and b such that

${\displaystyle \frac{7}{\sqrt{5}}=\frac{a}{b}}$

${\displaystyle \sqrt{5}=\frac{a}{7b}}$

We know that ${\displaystyle \sqrt{5}}$ is an irrational number

Here we see that ${\displaystyle \sqrt{5}}$ is a rational number which is a contradiction

Hence ${\displaystyle 7\sqrt{5}}$ is irrational