Question

If $x=3+2\sqrt{2}$, then find whether $x+\frac{1}{x}$ is rational or irrational.

• A. rational
• B. irrational
• C. can't say
• D. none of the above

Answer 1

The correct option is A rational

Given $x=3+2\sqrt{2}$

$x+\frac{1}{x}=3+2\sqrt{2}+\frac{1}{3+2\sqrt{2}}$

Rationalising second term,we get

$x+\frac{1}{x}=3+2\sqrt{2}+\frac{3-2\sqrt{2}}{(3+2\sqrt{2})(3-2\sqrt{2})}$

$=3+2\sqrt{2}+\frac{3-2\sqrt{2}}{9-8}$

$=3+2\sqrt{2}+3-2\sqrt{2}$

$=6$

which is a rational number.