Question

Let $P=(2+\sqrt{3}{)}^5$ and $f=P-\left[P\right]$,where [P] denotes the greatest integer function.Find the value of $\left(\frac{f^2}{1-f}\right)$

Answer 1

$P-(2+\sqrt{3}{)}^5$
$=(2+\sqrt{3}{)}^2(2+\sqrt{3}{)}^2(2+\sqrt{3})$
$=(7+4\sqrt{3})(7+4\sqrt{3})(2+\sqrt{3})$
$(97+56\sqrt{3})(2+\sqrt{3})$
$=(194+97\sqrt{3}+112\sqrt{3}+168)$
$362+208\sqrt{3}$
$\therefore =P=P-\left[P\right]=f=0.99$
$\therefore =9801$