6) $$^4\sqrt{\frac{4}{3}\sqrt[3]{\frac{3}{4}}\sqrt{\frac{4}{3}}}$$;

6) $$^4\sqrt{\frac{4}{3}\sqrt[3]{\frac{3}{4}}\sqrt{\frac{4}{3}}}$$;

$$^4\sqrt{\frac{4}{3}\sqrt[3]{\frac{3}{4}}\sqrt{\frac{4}{3}}} = \sqrt[4]{\frac{4}{3}\cdot \left(\frac{3}{4}\right)^{\frac{1}{3}}\cdot \left(\frac{4}{3}\right)^{\frac{1}{2}}} = \sqrt[4]{\left(\frac{4}{3}\right)^{\frac{6}{6}}\cdot \left(\frac{3}{4}\right)^{\frac{2}{6}}\cdot \left(\frac{4}{3}\right)^{\frac{3}{6}}} = \sqrt[4]{\left(\frac{4}{3}\right)^{\frac{9}{6}}\cdot \left(\frac{3}{4}\right)^{\frac{2}{6}}} = \sqrt[4]{\left(\frac{4}{3}\right)^{\frac{7}{6}}} = \left(\frac{4}{3}\right)^{\frac{7}{6*4}} = \left(\frac{4}{3}\right)^{\frac{7}{24}} = \sqrt[24]{\left(\frac{4}{3}\right)^7}$$

Ответ: $$\sqrt[24]{\left(\frac{4}{3}\right)^7}$$