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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