Prove that

Question:

Prove that $(2+3 \sqrt{5})$ is an irrational number, given that $\sqrt{5}$ is an irrational number.

 

Solution:

Let us assume that $(2+3 \sqrt{5})$ is a rational number.

Thus, $(2+3 \sqrt{5})$ can be represented in the form of $\frac{p}{q}$, where $p$ and $q$ are integers, $q \neq 0, p$ and $q$ are co-prime numbers.

$2+3 \sqrt{5}=\frac{p}{q}$

$\Rightarrow 3 \sqrt{5}=\frac{p}{q}-2$

$\Rightarrow 3 \sqrt{5}=\frac{p-2 q}{q}$

$\Rightarrow \sqrt{5}=\frac{p-2 q}{3 q}$

Since, $\frac{p-2 q}{3 q}$ is rational $\Rightarrow \sqrt{5}$ is rational

But, it is given that $\sqrt{5}$ is an irrational number.

Therefore, our assumption is wrong.

Hence, $2+3 \sqrt{5}$ is an irrational number.

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