Question

If $\frac{x+1}{x}=2$, find $\frac{x^3+1}{x^3}$

Answer 1

$x+\frac{1}{x}=2$
$\therefore x^3+{\frac{1}{x}}^3=?$
We have, $x+\frac{1}{x}=2$ ....(1)
$\Rightarrow {(x+\frac{1}{x})}^2=2^2$
$\Rightarrow x^2+\frac{1}{x^2}+2=4$
$\Rightarrow x^2+\frac{1}{x^2}=2$ ....(2)
(1) and (2)
$(x^2+{\frac{1}{x}}^2)(x+\frac{1}{x})=2\times 2$
$\Rightarrow x^3+x+\frac{1}{x}+\frac{1}{x^3}=4$
$\Rightarrow x^3+\frac{1}{x^3}+(x+\frac{1}{x})=4$
$\Rightarrow x^3+\frac{1}{x^3}=4-2=2$
$\therefore x^3+\frac{1}{x^3}=2$.