Prove that 3+5 is an irrational number. - Vidyadip

Question

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

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Answer

Let $3+\sqrt{5}$ is a rational number.
$3+\sqrt{5}=\frac{p}{q}, q \neq 0$
$3+\sqrt{5}=\frac{p}{q}$
$\Rightarrow \sqrt{5}=\frac{p}{q}-3$
$\Rightarrow \sqrt{5}=\frac{p-3 q}{q}$
Now in $\ce{RHS} \frac{p-3 q}{p}$ is rational
This shows that $\sqrt{5}$ is rational
But this contradict the fact that $\sqrt{5}$ is irrational,
This is because we assumed that $3+\sqrt{5}$ is a rational number.
$\therefore 3+\sqrt{5}$ is an irrational number.

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