To simplify the given expression, we first expand the numerator:

[tex] (\sqrt[4]{3} - \sqrt[4]{27})^{2} = (\sqrt[4]{3} - \sqrt[4]{27})(\sqrt[4]{3} - \sqrt[4]{27})[/tex]

Using the FOIL method (First, Outer, Inner, Last) to expand the expression, we get:

[tex] = (\sqrt[4]{3} \cdot \sqrt[4]{3}) + (\sqrt[4]{3} \cdot -\sqrt[4]{27}) + (-\sqrt[4]{27} \cdot \sqrt[4]{3}) + (-\sqrt[4]{27} \cdot -\sqrt[4]{27})[/tex]
[tex] = 3 - \sqrt[4]{81} - \sqrt[4]{81} + 27[/tex]
[tex] = 30 - 2\sqrt[4]{81}[/tex]
[tex] = 30 - 2(3)[/tex]
[tex] = 30 - 6[/tex]
[tex] = 24[/tex]

Now, we divide the result by the denominator:

[tex] 24 \div (6 - 4\sqrt{3})[/tex]
[tex] = 24 \div 2(3 - 2\sqrt{3})[/tex]
[tex] = 12 \div (3 - 2\sqrt{3})[/tex]

So, the final simplified expression is [tex] 12 \div (3 - 2\sqrt{3})[/tex].