Prove that \(3+2\sqrt{5}\) is irrational.
Proof:
Let \(3+2\sqrt{5}\) is .
be a rational number, where \(p\) and \(q\) are coprime and \(q\neq 0\)
Since, \(p\) and \(q\) are integers, \(\frac{\frac{p}{q}-3}{2}\) will also be rational.
Therefore, \(\sqrt{5}\) is rational.
This contradicts the fact that \(\sqrt{5}\) is irrational.
Hence, \(3+2\sqrt{5}\) is .
Answer variants:
\(3+2\sqrt{5}=\frac{p}{q}\)
rational
irrational
\(\sqrt{5}=\frac{\frac{p}{q}-3}{2}\)
\(2\sqrt{5}=\frac{p}{q}-3\)