Show that \(9+2\sqrt{5}\) is irrational.
Proof:
Let \(9+2\sqrt{5} =\frac{p}{q}\) be a number, where \(p\) and \(q\) are and \(q\neq 0\)
\(2\sqrt{5}=\frac{p}{q}-9\)
\(\sqrt{5}=\frac{p-9q}{2q}\)
Since, , \(\frac{p-9q}{2q}\) will also be .
Therefore, \(\sqrt{5}\) is rational.
This the fact that \(\sqrt{5}\) is .
Hence, \(9+2\sqrt{5}\) is irrational.