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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 written 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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