Question

Prove that \(6 - 4\sqrt{5}\) is an irrational number, given that \(\sqrt{5}\) is an irrational number.

Answer 1

Let \(x = 6 - 4\sqrt{5}\).

Assume for contradiction that \(x\) is rational, which means \(6 - 4\sqrt{5}\) is a rational number.

Rearranging:

\[ 4\sqrt{5} = 6 - x \]

\[ \sqrt{5} = \frac{6 - x}{4} \]

Since \(x\) is assumed to be rational, the right-hand side is rational, which implies that \(\sqrt{5}\) must be rational.

But \(\sqrt{5}\) is irrational, which is a contradiction. Therefore, \(6 - 4\sqrt{5}\) must be irrational.