Prove that $3 + 2 \sqrt{5}$ is irrational.

Prove that $3 + 2 \sqrt{5}$ is irrational.

Give Solution

Let us assume, for contradiction, that $ 3 + 2\sqrt{5} $ is rational.

That means we can write it as:

$ 3 + 2\sqrt{5} = \frac{a}{b} $, where $ a, b $ are integers and $ b \neq 0 $.

Rearranging:

$ 2\sqrt{5} = \frac{a}{b} - 3 $

$ 2\sqrt{5} = \frac{a - 3b}{b} $

Dividing by 2:

$ \sqrt{5} = \frac{a - 3b}{2b} $

Since $ a $ and $ b $ are integers, $ \frac{a - 3b}{2b} $ is a rational number.

However, $ \sqrt{5} $ is known to be irrational, which contradicts our assumption.

Therefore, $ 3 + 2\sqrt{5} $ is irrational.

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