If $x~=~2~+~{{2}^{\frac{2}{3}}}~~+~{{2}^{\frac{1}{3}}}$, then the value of ${{x}^{3}}~-~6{{x}^{2}}~+~6x$ is
Solution
$x~=~2~+~{{2}^{\frac{2}{3}}}~+~{{2}^{\frac{1}{3}}}$
$x-2~=~{{2}^{\frac{2}{3}}}+~{{2}^{\frac{1}{3}}}$
$x-2~=~{{2}^{\frac{1}{3}}}\left( ~{{2}^{\frac{1}{3}}}~+~1 \right)$ (i)
यहाँ, ${{x}^{3}}~-~6{{x}^{2}}~+~6x~=~{{\left( x-2 \right)}^{3}}~-~6x~+~8$ (ii)
तो, (i) के दोनों तरफ घन करें, और (ii) में डालें
हमें मिला,
${{\left( x-2 \right)}^{3}}~-~6x~+~8~=~2\times ~{{\left( \frac{21}{3}~+~1 \right)}^{3}}~~-~6\times ~(~2~+\frac{22}{3}~+\frac{21}{3})~+~8$
$=~2\times ~(~2~+~1~+~3.~{{2}^{\frac{2}{3}}}.1~+~3.1.~{{2}^{\frac{1}{3}}})~-~12~-~6.~{{2}^{\frac{2}{3}}}~-~6~.~{{2}^{\frac{1}{3}}}~+~8$
$=~6~-~12~+~8=2$