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# Prove that 3 +2 root 5 is irrational.

Q2  Prove that $3 + 2 \sqrt 5$  is irrational.

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Let us assume $3 + 2 \sqrt 5$ is rational.

This means $3 + 2 \sqrt 5$ can be wriiten in the form $\frac{p}{q}$ where p and q are co-prime integers.

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

As p and q are integers $\frac{p-3q}{2q}\\$ would be rational, this contradicts the fact that $\sqrt{5}$ is irrational. This contradiction arises because our initial assumption that $3 + 2 \sqrt 5$ is rational was wrong. Therefore $3 + 2 \sqrt 5$ is irrational.

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