Question

Given that \(\sqrt{5}\) is an irrational number, prove that \(2 + 3\sqrt{5}\) is an irrational number.

Hint:

If adding or multiplying an irrational number with a rational gives a rational, then the irrational number must become rational, which leads to contradiction.

Answer 1

Given:
\(\sqrt{5}\) is irrational.
We aim to prove \(2 + 3\sqrt{5}\) is irrational.

Step 1: Assume the opposite
Assume \(2 + 3\sqrt{5}\) is rational.
Let \(2 + 3\sqrt{5} = r\), where \(r\) is rational.

Step 2: Isolate \(\sqrt{5}\)
\[3\sqrt{5} = r - 2\] \[\sqrt{5} = \frac{r - 2}{3}\]

Step 3: Analyze the right side
Since \(r\) and 2 are rational, \(\frac{r - 2}{3}\) is also rational.
This means \(\sqrt{5}\) must be rational.

Step 4: The contradiction
This contradicts the given fact: \(\sqrt{5}\) is irrational.

Conclusion:
Our assumption is incorrect.
Therefore, \(2 + 3\sqrt{5}\) is irrational.

Final statement:
\[\boxed{2 + 3\sqrt{5} \text{ is irrational}}\]