Simplify. \( \left(\frac{4 z^{3}}{3 z^{7}}\right)^{-3} \) Write your answer using only positive exponents. 【I

To simplify \( \left(\frac{4 z^{3}}{3 z^{7}}\right)^{-3} \), first adjust the expression inside the parentheses. You can break it down as: \[ \frac{4 z^{3}}{3 z^{7}} = \frac{4}{3} \cdot \frac{z^{3}}{z^{7}} = \frac{4}{3} \cdot z^{3-7} = \frac{4}{3} \cdot z^{-4} \] Now raise that to the power of \(-3\): \[ \left( \frac{4}{3} \cdot z^{-4} \right)^{-3} = \left(\frac{4}{3}\right)^{-3} \cdot (z^{-4})^{-3} \] Calculating each part: \[ \left(\frac{4}{3}\right)^{-3} = \frac{1}{\left(\frac{4}{3}\right)^{3}} = \frac{1}{\frac{64}{27}} = \frac{27}{64} \] And for the exponent involving \( z \): \[ (z^{-4})^{-3} = z^{12} \] Combining these, we get: \[ \frac{27}{64} \cdot z^{12} = \frac{27 z^{12}}{64} \] Thus, the simplified answer using only positive exponents is: \[ \frac{27 z^{12}}{64} \]