\[ \left(\frac{a^{2} b^{-3} c^{-5}}{a^{5} b^{-7} c^{-8}}\right) \cdot\left(\frac{\sqrt{a^{3} b^{-9}}}{\sqrt[3]{c^{8}}}\right)=\sqrt[6]{\frac{c^{2}}{a^{9} b^{3}}} \] True False

Step 1 ：Simplify the left side of the equation: \(\left(\frac{a^{2} b^{-3} c^{-5}}{a^{5} b^{-7} c^{-8}}\right) \cdot\left(\frac{\sqrt{a^{3} b^{-9}}}{\sqrt[3]{c^{8}}}\right) = b^{4}c^{3}\sqrt{\frac{a^{3}}{b^{9}}}\div a^{3}\sqrt[3]{c^{8}}\)

Step 2 ：Simplify the right side of the equation: \(\sqrt[6]{\frac{c^{2}}{a^{9} b^{3}}} = \sqrt{\frac{c^{2}}{a^{9} b^{3}}}\)

Step 3 ：Compare the simplified left and right sides of the equation: \(b^{4}c^{3}\sqrt{\frac{a^{3}}{b^{9}}}\div a^{3}\sqrt[3]{c^{8}}\) is not equal to \(\sqrt{\frac{c^{2}}{a^{9} b^{3}}}\)

Step 4 ：Final Answer: The given equation is \(\boxed{\text{False}}\)