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| 1. |
Given that root 2 is an irrational number ,then prove that 5+3root2 is an irrational number . |
| Answer» To Prove: 5+3{tex}\\sqrt{2}{/tex}\xa0is irrational numberProof: If possible let us assume 5 + 3{tex}\\sqrt{2}{/tex}\xa0is a rational number.{tex}\\implies {/tex}\xa05 + 3{tex}\\sqrt{2}{/tex}\xa0=\xa0{tex}\\frac{p}{q}{/tex}\xa0where q\xa0{tex}\\ne{/tex}\xa00 and p and q are coprime integers.{tex}\\implies{/tex}{tex}3\\sqrt{2}=\\frac{p}{q}-5 {/tex}{tex}\\implies{/tex}{tex}3\\sqrt{2}=\\frac{p-5q}{q}{/tex}{tex}\\implies\\sqrt{2}=\\frac{p-5q}{3q}{/tex}{tex}\\implies\\sqrt{2}=\\frac{integer}{integer}{/tex}{tex}\\Rightarrow{/tex}\xa0{tex}\\sqrt{2}{/tex} is\xa0a rational number.This contradicts the given fact that\xa0{tex}\\sqrt{2}{/tex}\xa0is irrational.Hence 5 + 3{tex}\\sqrt{2}{/tex} is an irrational number. | |