Question

Show that $7-\sqrt{5}$ is irrational given that $\sqrt{5}$ is irrational .

Answer 1

Let $7-\sqrt{5}$ a rational number.

As Rational-Rational is also a rational number.

$\Rightarrow \left(7\right)-(7-\sqrt{5})=\sqrt{5}$ a rational number which is false

$\Rightarrow 7-\sqrt{5}$ is an irrational numbers.