It is given that x=√13+2√3. Then,
1x=1√13+2√3
x−1x=(√13+2√3)−1(√13+2√3)
=(√13+2√3)2−1(√13+2√3)
=13+12+4√39−1√13+2√3
=24−4√39√13+2√3
Rationalizing the denominator,
24+4√39√13+2√3×√13−2√3√13−2√3=(24+4√39)(√13−2√3)(√13+2√3)(√13−2√3)
=24√13−48√3+4√13×3×√13−8√13×3×√3(√13)2−(2√3)2
=24√13−48√3+52√3−24√1313−12
=4√31
=4√3