Prove that 2 root 3 -1 is an irrational number

1.	Prove that 2 root 3 -1 is an irrational number
Answer» Let us assume that {tex}2\\sqrt 3 - 1{/tex}\xa0is a rational.\xa0numberThen, there exist positive co-primes a and b such that{tex}2\\sqrt 3 - 1 = \\frac{a}{b}{/tex}{tex}2\\sqrt 3 = \\frac{a}{b} + 1{/tex}{tex}\\begin{array}{l}2\\sqrt3=\\frac{\\mathrm a+\\mathrm b}{\\mathrm b}\\\\\\end{array}{/tex}{tex}\\sqrt 3 = \\frac{{a + b}}{{2b}}{/tex}Here\xa0{tex}\\begin{array}{l}\\frac{\\mathrm a+\\mathrm b}{\\mathrm b}\\\\\\end{array}{/tex}\xa0is a rational number ,so {tex}\\sqrt3{/tex}\xa0is a rational numberThis contradicts the fact that {tex}\\sqrt 3{/tex}\xa0is an irrational numberHence {tex}2\\sqrt 3 - 1{/tex}\xa0is irrational

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