Simplify: \(\left(\frac{\sqrt{2}}{\sqrt{3}}\right)^5 \left(\frac{6}{7}\right)^2\)
Solution
\[ \left(\frac{\sqrt{2}}{\sqrt{3}}\right)^5 \left(\frac{6}{7}\right)^2 \]
\[ = \frac{(\sqrt{2})^5}{(\sqrt{3})^5} \cdot \frac{6^2}{7^2} \]
\[ = \frac{2^{5/2}}{3^{5/2}} \cdot \frac{36}{49} \]
\[ = \frac{4\sqrt{2}}{9\sqrt{3}} \cdot \frac{36}{49} \]
\[ = \frac{144\sqrt{2}}{441\sqrt{3}} \]
\[ = \frac{48\sqrt{2}}{147\sqrt{3}} \]
\[ = \frac{48\sqrt{6}}{441} \]
Final Answer:
\[ \boxed{\frac{48\sqrt{6}}{441}} \]