Given that is irrational, prove that is irrational.

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Solution

Q.
Given that $\sqrt{3}$ is irrational, prove that $5 + 2 \sqrt{3}$ is irrational.

Ans.
Let us assume $5 + 2 \sqrt{3}$ is rational, then it must be in the form of p/q where p and q are coprime integers and $q \neq 0$

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

So $\sqrt{3} = \frac{p - 5 q}{2 q}$ ...(i)

Since p, q, 5 and 2 are integers and $q \neq 0,$

RHS of equation (i) is rational.

But LHS of (i) is $\sqrt{3}$ which is irrational. This is not possible.

This contradiction has arisen due to our wrong assumption that $5 + 2 \sqrt{3}$ is rational.

So, $5 + 2 \sqrt{3}$ is irrational.

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