Question:

Prove that 6456 - 4\sqrt{5} is an irrational number, given that 5\sqrt{5} is an irrational number.

Updated On: Dec 12, 2024
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Solution and Explanation

Let x=645x = 6 - 4\sqrt{5}.

Assume for contradiction that xx is rational, which means 6456 - 4\sqrt{5} is a rational number.

Rearranging:

45=6x 4\sqrt{5} = 6 - x

5=6x4 \sqrt{5} = \frac{6 - x}{4}

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

But 5\sqrt{5} is irrational, which is a contradiction. Therefore, 6456 - 4\sqrt{5} must be irrational.

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