Question

question

Answer 1

\(\large \color{black}{\begin{align} &\normalsize \text{If}\ \ \large p=2^{2/3}+2^{1/3} \hspace{.33em}\\~\\
& \normalsize \text{then which of the following is true ?} \hspace{.33em}\\~\\
&a.)\ p^3-6p-6=0 \hspace{.33em}\\~\\
&b.)\ p^3-6p+6=0 \hspace{.33em}\\~\\
&c.)\ p^3+6p-6=0 \hspace{.33em}\\~\\
&a.)\ p^3+6p+6=0 \hspace{.33em}\\~\\
\end{align}}\)

Answer 2

a.) by trial and error using Mathematica v9

Answer 3

if $$p=2^{2/3}+2^{1/3}$$ then notice $$p^3=(2^{2/3}+2^{1/3})^3=2^2+2+3(2^{4/3}\cdot2^{1/3}+2^{2/3}\cdot2^{2/3})\\p^3=6+3\cdot (2^{5/3}+2^{4/3})=6+6(2^{2/3}+2^{1/3})\\p^3=6+6p\\p^3-6p-6=0$$