Question

Prove that \(2 + 3\sqrt{5}\) is an irrational number given that \(\sqrt{5}\) is an irrational number.

Hint:

In such "prove that" questions, always isolate the root term on one side. If the other side consists entirely of known rational operations (addition, subtraction, multiplication, division) on integers, that side is rational, leading to the contradiction.

Answer 1

Step 1: Assume the Opposite
Assume 2 + 3√5 is rational.

Then it can be written as:
2 + 3√5 = a/b
where a and b are integers and b ≠ 0.

Step 2: Isolate √5
3√5 = a/b − 2

Take LCM:
3√5 = (a − 2b)/b

Divide both sides by 3:
√5 = (a − 2b)/(3b)

Step 3: Observe the Result
Since a and b are integers,
(a − 2b) and 3b are also integers.

So (a − 2b)/(3b) is a rational number.

This implies √5 is rational.

But we know √5 is irrational.

This is a contradiction.

Step 4: Conclusion
Our assumption is wrong.

Final Answer:
Therefore, 2 + 3√5 is irrational.