\( \begin{array}{l} \text { Andrea }\\ \checkmark 22\\ 23\\ 24\\ 25\\ 26\\ \times 27\\ 28\\ 29\\ 30\\ \text { Espanol }\\ 33\\ \text { Divide the rational expressions. }\\ \frac{z}{z-9} \div \frac{z^{3}+5 z^{2}-36 z}{z^{2}-81}= \end{array} \)

To divide the given rational expressions: \[ \frac{z}{z - 9} \div \frac{z^{3} + 5z^{2} - 36z}{z^{2} - 81} \] **Follow these steps to simplify:** ### 1. Rewrite the Division as Multiplication by the Reciprocal Dividing by a fraction is the same as multiplying by its reciprocal. So, \[ \frac{z}{z - 9} \div \frac{z^{3} + 5z^{2} - 36z}{z^{2} - 81} = \frac{z}{z - 9} \times \frac{z^{2} - 81}{z^{3} + 5z^{2} - 36z} \] ### 2. Factor All Possible Expressions Let's factor the numerator and the denominator wherever possible. - **Factor \( z^{2} - 81 \):** This is a difference of squares. \[ z^{2} - 81 = (z - 9)(z + 9) \] - **Factor \( z^{3} + 5z^{2} - 36z \):** Factor out the greatest common factor (GCF), which is \( z \): \[ z^{3} + 5z^{2} - 36z = z(z^{2} + 5z - 36) \] Now, factor the quadratic \( z^{2} + 5z - 36 \): \[ z^{2} + 5z - 36 = (z + 9)(z - 4) \] So, \[ z^{3} + 5z^{2} - 36z = z(z + 9)(z - 4) \] ### 3. Substitute the Factored Forms Back into the Expression \[ \frac{z}{z - 9} \times \frac{(z - 9)(z + 9)}{z(z + 9)(z - 4)} \] ### 4. Cancel Common Factors Identify and cancel out the common terms in the numerator and the denominator. - **Cancel \( z \):** \[ \frac{\cancel{z}}{z - 9} \times \frac{(z - 9)(z + 9)}{\cancel{z}(z + 9)(z - 4)} = \frac{(z - 9)(z + 9)}{(z - 9)(z + 9)(z - 4)} \] - **Cancel \( z - 9 \):** \[ \frac{\cancel{(z - 9)}(z + 9)}{\cancel{(z - 9)}(z + 9)(z - 4)} = \frac{z + 9}{(z + 9)(z - 4)} \] - **Cancel \( z + 9 \):** \[ \frac{\cancel{z + 9}}{\cancel{z + 9}(z - 4)} = \frac{1}{z - 4} \] ### 5. Final Simplified Expression \[ \frac{z}{z - 9} \div \frac{z^{3} + 5z^{2} - 36z}{z^{2} - 81} = \frac{1}{z - 4} \] **Answer:** \[ \frac{1}{z - 4} \]