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InterviewSolution
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Power of 3√6 is irrational |
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Answer» Suppose\xa0{tex}\\sqrt [ 3 ] { 6 }{/tex}\xa0be rational number and\xa0{tex}\\sqrt [ 3 ] { 6 } = \\frac { a } { b }{/tex}\xa0where {tex}a\\ and\\ b{/tex} are co-prime and\xa0{tex}b\\ne0{/tex}{tex}\\Rightarrow ( \\sqrt [ 3 ] { 6 } ) ^ { 3 } = \\frac { a ^ { 3 } } { b ^ { 3 } }{/tex}{tex}\\Rightarrow 6 = \\frac { a ^ { 3 } } { b ^ { 3 } } {/tex}{tex}\\Rightarrow 6 . b ^ { 3 } = a ^ { 3 }{/tex}{tex}\\Rightarrow {/tex}\xa0{tex}a^3{/tex}\xa0is divisible by {tex}6{/tex}\xa0{tex}\\Rightarrow{/tex}\xa0{tex}a{/tex} is divisible by {tex}6{/tex}.Let {tex}a = 6c{/tex}{tex}6b^3\xa0= (6c)^3{/tex}{tex}\\Rightarrow \\quad b ^ { 3 } = 36 c ^ { 3 }{/tex}{tex}\\Rightarrow {/tex}\xa0{tex}b^3{/tex}\xa0is divisible by {tex}6{/tex}\xa0{tex}\\Rightarrow {/tex}\xa0{tex}b{/tex} is divisible by {tex}6{/tex}.{tex}\\Rightarrow {/tex}\xa0{tex}a\\ and\\ b{/tex} have a common factor i.e, {tex}6{/tex}{tex}\\Rightarrow {/tex}\xa0{tex}a\\ and\\ b{/tex} are not co-prime which is a contradiction{tex}\\therefore \\sqrt [ 3 ] { 6 }{/tex}\xa0is an irratonal. Is irrational |
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