Prove that \(8+3\sqrt{11}\) is an irrational.
Proof:
Assume \(8+3\sqrt{11}\) is .
If \(8+3\sqrt{11}\) is rational, it can be written as a fraction
\(\sqrt{11}=\)
Here, both and \(3q\) are integers.
Therefore, is a rational number.
This implies that \(\sqrt{11}\) is a number.
This contradicts the fact that \(\sqrt{11}\) is .
Therefore, \(8+3\sqrt{11}\) is an .
Answer variants:
\(4-3q\)
rational
\(7p-q\)
\(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(= 0\)
\(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q\ne 0\)
\(p-8q\)
\(\frac{p-8q}{3q}\)
\(\frac{q+8p}{3+p}\)
irrational