Question

If X= 3+2 root 3 then what is the value of X-1/X​

Answer 1

Answer:

$x = 3 + 2 \sqrt{3} \\ \\ \frac{1}{x} = \frac{1}{3 + 2 \sqrt{3} } \times \frac{3 - 2 \sqrt{3} }{3 - 2 \sqrt{3} } \\ \\ \frac{1}{x} = \frac{3 - 2 \sqrt{3} }{(3) {}^{2} - (2 \sqrt{3}) {}^{2} } \\ \\ \frac{1}{x} = \frac{3 - 2 \sqrt{3} }{9 - 12} \\ \\ \frac{1}{x} = \frac{-3 +2 \sqrt{2}} {2}$

Now,

$= x - \frac{1}{x} \\ \\ = 3 + 2 \sqrt{3} - \frac{-3 +2 \sqrt{2}} {2} \\ \\ = \frac{2(3+2 \sqrt {3}) + 3 - 2\sqrt {3}}{2} \\ \\ = \frac{6+4 \sqrt{3} + 3 - 2 \sqrt {3}} {2} \\ \\ = \frac{9 + 2 \sqrt {3}} {2}$

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Answer 2

given,

x = 3+2√3

x-1/x

= (x²-1)/x

= [(3+2√3)²-1]/3+2√3

= [9+12+12√3-1]/3+2√3

= (20+12√3)/3+2√3

on rationalizing denominator with 3-2√3

= [(20+12√3)(3-2√3)]/(-4)

= (60-40√3+36√3-72)/(-4)

= (-12-4√3)/(-4)

= 3+√3