Question

Show that 7+2√5 is irrational

Answer 1

$\sf We\ know\ that\ \sqrt{5} \ is\ irrational.\\\\Let\ us\ assume\ that\ 7 + 2\sqrt{5}\ is\ rational.\\\\7 + 2\sqrt{5}\ = \ \frac{a}{b},\ where\ a\ and\ b\ are\ co-prime\ integers\ and\ b\neq 0.\\\\7 + 2\sqrt{5} = \frac{a}{b}\\ \\\frac{7b\ -\ a}{b} = -2\sqrt{5}\\ \\\frac{7b\ -\ a}{-2b} = \sqrt{5}\\\\Since\ a\ and\ b\ are\ integers,\ \frac{7b\ -\ a}{-2b}\ is\ a\ rational\ number.\\\\This\ implies\ that\ \sqrt{5}\ is\ rational\ too!$

$\sf This\ contradicts\ the\ fact\ that\ \sqrt{5}\ is\ irrational.\\\\Therefore,\ our\ asumption\ is\ wrong.\\\\Hence, 7 + 2\sqrt{5}\ is\ irrational.$

Answer 2

Answer:

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