Question

If $x^3-\left[x\right]=3$, where, $\left[x\right]$ denotes the greatest integer less than or equal to the number $x$, then $x^3$ is equal to

Answer 1

$\because x=\left[x\right]+f,0\le f<1$
And given equation is $x^3-\left[x\right]=3$
$\Rightarrow x^3-(x-f)=3$
$\Rightarrow i.e.x^3-x=3-f$
where it follows that
$2<x^3-x\le 3$
Now,
$1<x<2$. (If we consider the range of values of $x^3-x$ )
Hence, $\left[x\right]=1$
Hence, $x^3=4$