Question

Solve: $\left[{\left(\sqrt[6]{67}\right)}^2+{\left(\sqrt[6]{67}\right)}^{-2}\right]\left[{\left(\sqrt[6]{67}\right)}^4+{\left(\sqrt[6]{67}\right)}^{-4}\right]$

Answer 1

Consider the given expression.

$[{\left(\sqrt[6]{67}\right)}^2+{\left(\sqrt[6]{67}\right)}^{-2}][{\left(\sqrt[6]{67}\right)}^4+{\left(\sqrt[6]{67}\right)}^{-4}]$

$\Rightarrow [{\left(\sqrt[6]{67}\right)}^2+\frac{1}{{\left(\sqrt[6]{67}\right)}^2}][{\left(\sqrt[6]{67}\right)}^4+\frac{1}{\left(\sqrt[6]{67}\right)4}]$

$\Rightarrow [{\left(7\right)}^{2/6}+\frac{1}{{\left(7\right)}^{2/6}}][{\left(7\right)}^{4/6}+\frac{1}{{\left(7\right)}^{4/6}}]$

$\Rightarrow [{\left(7\right)}^{1/3}+\frac{1}{{\left(7\right)}^{1/3}}][{\left(7\right)}^{2/3}+\frac{1}{{\left(7\right)}^{2/3}}]$

$\Rightarrow \left(\frac{{\left(7\right)}^{2/3}+1}{{\left(7\right)}^{1/3}}\right)\left(\frac{{\left(7\right)}^{4/3}+1}{{\left(7\right)}^{2/3}}\right)$

$\Rightarrow \left(\frac{{\left(7\right)}^{2/3}+1}{{\left(7\right)}^{1/3}}\right)\left(\frac{{\left(7\right)}^{4/3}+{\left(7\right)}^{2/3}}{{\left(7\right)}^{2/3}}\right)$

$\Rightarrow ({\left(7\right)}^{2/3}+1)\left(\frac{{\left(7\right)}^{4/3}+{\left(7\right)}^{2/3}}{7}\right)$

$\Rightarrow ({\left(7\right)}^{2/3}+1)\left(\frac{{\left(7\right)}^{1/3}\times 7+7\times {\left(7\right)}^{-1/3}}{7}\right)$

$\Rightarrow ({\left(7\right)}^{2/3}+1)({\left(7\right)}^{1/3}+{\left(7\right)}^{-1/3})$

$\Rightarrow ({\left(7\right)}^{2/3}+1)\left(\frac{{\left(7\right)}^{2/3}+1}{7^{1/3}}\right)$

$\Rightarrow \frac{{({\left(7\right)}^{2/3}+1)}^2}{7^{1/3}}$

Hence, this is the answer.