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Show that \frac{2+3\sqrt{2}}{7}is not a rational number, given that \sqrt{2} is an irrational number.

 

 
 
 
 
 

Answers (1)

Given \sqrt{2} is an irrational number
To prove \rightarrow \frac{2+3\sqrt{2}}{7} is not rational number
Proof \rightarrow Let's assume \frac{2+3\sqrt{2}}{7} is rational number
Property of Rational no \rightarrow multiplication of also rational no = rational no
\Rightarrow \left ( \frac{2+3\sqrt{2}}{7} \right )\times 7= 2+3\sqrt{2} is also rational (\therefore 7 is rational)
Property of Rational no \rightarrow difference of also rational no = rational no
\Rightarrow 2+3\sqrt{2}-2= 3\sqrt{2} is also rational  (\therefore 2 is rational)
Property of Rational no \rightarrow division of a rational no by other rational number reacts into rational no
\Rightarrow \frac{3\sqrt{2}}{3}= \sqrt{2} is rational (\therefore 3 is rational)
But as we already know \sqrt{2} is irrational no hence by contradiction the no \frac{2+3\sqrt{2}}{7} is not rational

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Safeer PP

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