Question

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

Answer 1

Let $\sqrt{5}+\sqrt{3}$ be any rational number $x$
$\Rightarrow x=\sqrt{5}+\sqrt{3}$

squaring both sides
$\Rightarrow x^2=(\sqrt{5}+\sqrt{3}{)}^2$
$\Rightarrow x^2=3+5+2\sqrt{1}5$
$\Rightarrow x^2=8+2\sqrt{1}5$

$\Rightarrow \frac{x^2-8}{2}=\sqrt{1}5$

As $x$ is a rational number so $x^2$ is also a rational number.
Since we know 8 and 2 are rational numbers, $\sqrt{1}5$ must also be a rational number as quotient of two rational numbers is rational.
But $\sqrt{1}5$ is an irrational number so we arrive at a contradiction.
This shows that our assumption was wrong.

Also, ​​​​​$\sqrt{3}$ and $\sqrt{5}$ are irrational numbers and we know that the sum of two irrational numbers is also irrational.

$\therefore \sqrt{5}+\sqrt{3}$ is not a rational number.