Question

If x=3+2root 2 then check whether x+1/x is a rational or irrational

Answer 1

X = 3+2 root 2

1/X = 1/3+ 2 root 2

Rationalizing the denominator 3 + 2 root 2 we get,

1/X = 1/ 3 + 2 root 2 × 3 - 2 root 2/3 - 2root2

1/X = ( 3 - 2 root 2)/ ( 3 + 2 root 2) ( 3 - 2root2)

1/X = ( 3 - 2root 2) / (3)² - ( 2 root2)²

1/X = ( 3 - 2 root 2) / 9 -8

1/X = ( 3 - 2 root 2)

Therefore,

X + 1/X

=> ( 3 + 2 root 2 ) + ( 3 - 2 root 2)

=> ( 3 2 root 2)+ ( 2 root 2)

=> 6 which is rational.

Hence,

X + 1/X is rational

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Answer 2

$\textbf {Your answer !!}$

$x = 3 + 2 \sqrt{2} \\ \\ lets \: find \: \frac{1}{x} \\ \\ = > \frac{1}{x} = \frac{1}{3 + 2 \sqrt{2} } \\ \\ = > \frac{1}{x} = \frac{1}{3 + 2 \sqrt{2} } \times \frac{3 - 2 \sqrt{2} }{3 - 2 \sqrt{2} } \\ \\ = > \frac{1}{x} = \frac{3 - 2 \sqrt{2} }{ {(3)}^{2} - {(2 \sqrt{2} )}^{2} } \\ \\ = > \frac{1}{x} = \frac{3 - 2 \sqrt{2} }{9 - 8} \\ \\ = > \frac{1}{x} = \frac{3 - 2 \sqrt{2} }{1} \\ \\ = > \frac{1}{x} = 3 - 2 \sqrt{2} \\ \\ now \: x + \frac{1}{x} \\ \\ put \\ = > \: x = 3 + 2 \sqrt{2} \\ and \\ = > \frac{1}{x} = 3 - 2 \sqrt{2} \\ \\ = > ........................ > > ........................ \\ \\ = > x + \frac{1}{x} = 3 + 2 \sqrt{2} + 3 - 2 \sqrt{2} \\ \\ = > x + \frac{1}{x} = 3 + 3 + 2 \sqrt{2} - 2 \sqrt{2} \\ \\ = > x + \frac{1}{ x } = 6$

$\textbf {Hence, It Is Rational !!}$

$\text { Thanka !!}$