Show that

Question:

Show that $5-2 \sqrt{3}$ is an irrational number.

Solution:

Let us assume that $5-2 \sqrt{3}$ is rational .Then, there exist positive co primes $a$ and $b$ such that

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

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

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

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

This contradicts the fact that $\sqrt{3}$ is an irrational number

Hence $5-2 \sqrt{3}$ is irrational

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