Question

If $x=2+\sqrt{3}$, find the value of $x^3+1/x^3$.

Answer 1

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

Now,

$\frac{1}{x}$

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

$=\frac{2-\sqrt{3}}{2^2-{\sqrt{3}}^2}$

$=2-\sqrt{3}$.

Then, $x+\frac{1}{x}=4$......(1).

Now,

$x^3+\frac{1}{x^3}$

$={(x+\frac{1}{x})}^3-3.x.\frac{1}{x}.(x+\frac{1}{x})$

$=4^3-3.4$

$=64-12$

$=52$.