1. **State the problem:** Simplify the monomial expression $$\frac{18 a^{3} b^{-2} \cdot 5 a b^{2}}{9 a^{4} b \cdot (- a b^{3})}$$ and express the final answer using only positive exponents. 2. **Write the expression clearly:** $$\frac{18 a^{3} b^{-2} \times 5 a b^{2}}{9 a^{4} b \times (- a b^{3})}$$ 3. **Multiply the numerators and denominators separately:** Numerator: $$18 \times 5 \times a^{3} \times a \times b^{-2} \times b^{2} = 90 a^{4} b^{0}$$ Since $$b^{-2} \times b^{2} = b^{-2+2} = b^{0} = 1$$ Denominator: $$9 \times (-1) \times a^{4} \times a \times b \times b^{3} = -9 a^{5} b^{4}$$ 4. **Rewrite the fraction:** $$\frac{90 a^{4}}{-9 a^{5} b^{4}}$$ 5. **Simplify the coefficients:** $$\frac{\cancel{90}^{10}}{\cancel{-9}^{-1}} = -10$$ 6. **Simplify the variables:** $$\frac{a^{4}}{a^{5}} = a^{4-5} = a^{-1}$$ 7. **Combine all parts:** $$-10 a^{-1} b^{-4}$$ 8. **Express with positive exponents:** $$- \frac{10}{a b^{4}}$$ **Final answer:** $$- \frac{10}{a b^{4}}$$