[tex] \sqrt[4]{2} \times \sqrt[4]{3} \div \sqrt[4]{486} + \sqrt[3]{27 \times {2}^{6} } [/tex]

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We first simplify each term inside the square roots and cube roots:

[tex] \sqrt[4]{2} = 2^{\frac{1}{4}} [/tex][tex] \sqrt[4]{3} = 3^{\frac{1}{4}} [/tex][tex] \sqrt[4]{486} = 486^{\frac{1}{4}} = (2 \times 3^2)^{\frac{1}{4}} = 2^{\frac{1}{4}} \times 3^{\frac{2}{4}} = 2^{\frac{1}{4}} \times 3^{\frac{1}{2}} [/tex][tex] \sqrt[3]{27 \times 2^6} = (27 \times 2^6)^{\frac{1}{3}} = (3^3 \times 2^6)^{\frac{1}{3}} = 3 \times 2^2 [/tex]

Now we substitute these simplified forms back into the expression:

[tex] 2^{\frac{1}{4}} \times 3^{\frac{1}{4}} \div (2^{\frac{1}{4}} \times 3^{\frac{1}{2}}) + 3 \times 2^2 [/tex]

Now, when we divide [tex] 2^{\frac{1}{4}} [/tex] by [tex] 2^{\frac{1}{4}} [/tex], we get 1. So, the equation becomes:

[tex] 1 \times 3^{\frac{1}{4}} \div 3^{\frac{1}{2}} + 3 \times 2^2 [/tex]

This simplifies to:

[tex] 3^{\frac{-1}{4}} + 3 \times 4 [/tex]
[tex] = \frac{1}{3^{\frac{1}{4}}} + 12 [/tex]
[tex] = \frac{1}{\sqrt[4]{3}} + 12 [/tex]