Given that root 3 is an irrational number prove that 2+3 root 3 is an irrational number.

Given that root 3 is an irrational number prove that 2+3 root 3 is an

1.	Given that root 3 is an irrational number prove that 2+3 root 3 is an irrational number.
Answer» To Prove: 2+{tex}\\sqrt3{/tex}\xa0is an irratinal number.Given:\xa0{tex}\\sqrt3{/tex}\xa0is irrational number.Proof: Let 2 +\xa0{tex}\\sqrt{3}{/tex}\xa0be a rational number.{tex}\\Rightarrow{/tex}\xa02 +\xa0{tex}\\sqrt{3}{/tex}\xa0=\xa0{tex}\\frac{p}{q}{/tex}, p, q\xa0{tex}\\in{/tex}\xa0I, q\xa0{tex}\\ne{/tex}\xa00\xa0{tex}\\Rightarrow{/tex}\xa0{tex}\\sqrt{3}{/tex}\xa0=\xa0{tex}\\frac{p}{q}{/tex}\xa0- 2 =\xa0{tex}\\frac{p - 2q}{q}{/tex} =\xa0{tex}\\frac{integer}{integer}{/tex}\xa0{tex}\\implies{/tex}{tex}\\sqrt{3}{/tex}\xa0is rational number\xa0{tex}\\Rightarrow{/tex} which is a contradiction to the fact that\xa0{tex}\\sqrt{3}{/tex}\xa0is a rational\xa0hence 2 +\xa0{tex}\\sqrt{3}{/tex} is irrational number.