Question

If $x=2+2^{2/3}+2^{1/3}$, then find the value of $x^3-6x^2+6x$

Answer 1

Answer: 2.00

$x=2+2^{2/3}+2^{1/3}$
$x^3-6x^2+6x$
$x-2=\left(2^{2/3}+2^{1/3}\right)...(i)$
making whole cube
$(x-2{)}^3={\left(2^{2/3}+2^{1/3}\right)}^3$
$x^3-8-6x(x-2)$
$=4+2+3\cdot \left[2^{2/3}\cdot 2^{1/3}\right]$
$\left(2^{2/3}+2^{1/3}\right)$
$x^3-8-6x^2+12x$
$=6+6(x-2)$
$x^3-8-6x^2+12x$
$=6+6x-12$
$x^3-6x^2+6x=2$