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Prove that 2+3\sqrt{3} is an irrational number when it is given that \sqrt{3} is an irrational number.

 

 

 
 
 
 
 

Answers (1)

Given \rightarrow \sqrt{3} is an irrational number
To Prove \rightarrow 2+3\sqrt{3} is irrational no
Proof \rightarrow Let's assume 2+3\sqrt{3} is rational no
we know that the 2 is rational no
Property of Rational no \rightarrow difference between two rational no's is always rational
\Rightarrow 2+3\sqrt{3}-2= 3\sqrt{3}
By the above property 3\sqrt{3} is also a rational no
Property of Rational no \rightarrow product of two rational no's is also a rational
we know \frac{1}{3} is rational no
\Rightarrow 3\sqrt{3}\times \frac{1}{3}= \sqrt{3}
By the above property \sqrt{3} is also rational but which is a contradiction
Hence by contradiction, it is clear that 2+3\sqrt{3} is an irrational no.

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Safeer PP

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