Question

Given that $\sqrt{3}and\sqrt{5}$ are irrational numbers, prove that $\sqrt{3}+\sqrt{5}$ is an irrational number.

Answer 1

$\sqrt{3}$ & $\sqrt{5}$ are irrational [given]

To prove: $\sqrt{3}+\sqrt{5}$ is irrational

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

Now $\sqrt{3}+\sqrt{5}=\frac{a}{b},$ where a & b are coprime

integers as $b\ne o$

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

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

$\Rightarrow \frac{a-\sqrt{5}b}{b}$ is a form of $\frac{a}{b}$ then

it is rational number

If $\frac{a-\sqrt{5}b}{b}$ is a rational number

then $\sqrt{3}$ is also rational numbers

which is not true

$\therefore$ our assumption is false

$\sqrt{3}+\sqrt{5}$ is an irrational number