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Pove that 7 is irrational number |
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Answer» \xa0let us assume that\xa0{tex}\\sqrt 7{/tex}\xa0be a rational number.{tex}\\sqrt { 7 } = \\frac { a } { b }{/tex}, where a and b are integers and co-primes and b{tex} \\neq{/tex}0Squaring both sides, we have{tex}\\frac { a ^ { 2 } } { b ^ { 2 } } = 3{/tex}or,\xa0{tex}a ^ { 2 } = 7 b ^ { 2 }{/tex}--------(i)a2\xa0is divisible by 7.Hence a is divisible by 7..........(ii)Let a = 7c ( where c is any integer)squaring on both sides we get(7c)2\xa0= 7b249c2\xa0= 7b2b2\xa0= 7c2so b2\xa0is divisible by 7hence, b is divisible by 7..........(iii)From equation(ii) and (iii), we have7 is a factor of a and b which is contradicting the fact that a and b are co-primes.Thus, our assumption that\xa0{tex}\\sqrt 7{/tex} is rational number is wrong.Hence,\xa0{tex}\\sqrt 7{/tex}\xa0is an irrational number. Yes |
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