1. **Problem 4:** Simplify $ \frac{2w^{-3}p}{4w^{3}p^{-2}} $ to an equivalent expression with exponents and fewest factors. 2. Use the quotient rule for exponents: $ \frac{a^{m}}{a^{n}} = a^{m-n} $. 3. Simplify coefficients: $ \frac{2}{4} = \frac{1}{2} $. 4. Simplify $ w $ terms: $ w^{-3 - 3} = w^{-6} $. 5. Simplify $ p $ terms: $ p^{1 - (-2)} = p^{3} $. 6. Final simplified expression: $$ \frac{1}{2} w^{-6} p^{3} $$ --- 1. **Problem 5:** Simplify $ \frac{10k^{6}j^{9}}{5k^{8}j^{-2}} $. 2. Simplify coefficients: $ \frac{10}{5} = 2 $. 3. Simplify $ k $ terms: $ k^{6 - 8} = k^{-2} $. 4. Simplify $ j $ terms: $ j^{9 - (-2)} = j^{11} $. 5. Final simplified expression: $$ 2k^{-2}j^{11} $$ --- 1. **Problem 6:** Simplify $$ \left( \frac{3f^{4}g^{-5}}{4f^{-2}g^{8}} \right)^{2} $$. 2. Simplify inside the parentheses first. 3. Coefficients: $ \frac{3}{4} $. 4. $ f $ terms: $ f^{4 - (-2)} = f^{6} $. 5. $ g $ terms: $ g^{-5 - 8} = g^{-13} $. 6. Expression inside parentheses: $$ \frac{3}{4} f^{6} g^{-13} $$. 7. Now square everything: $$ \left( \frac{3}{4} \right)^{2} = \frac{9}{16} $$ $$ (f^{6})^{2} = f^{12} $$ $$ (g^{-13})^{2} = g^{-26} $$ 8. Final expression: $$ \frac{9}{16} f^{12} g^{-26} $$ --- 1. **Problem 7:** Simplify $ (m^{-4} \cdot m^{0})^{-6} $. 2. Use product rule: $ m^{-4 + 0} = m^{-4} $. 3. Expression becomes $ (m^{-4})^{-6} $. 4. Power of a power: $ m^{-4 \times -6} = m^{24} $. 5. Final expression: $$ m^{24} $$ --- 1. **Problem 8:** Simplify $ \frac{c^{4}}{12c^{14} \cdot (-3c^{5})} $. 2. Multiply denominator coefficients: $ 12 \times (-3) = -36 $. 3. Multiply denominator variables: $ c^{14} \cdot c^{5} = c^{19} $. 4. Expression becomes $ \frac{c^{4}}{-36 c^{19}} $. 5. Simplify $ c $ terms: $ c^{4 - 19} = c^{-15} $. 6. Final expression: $$ \frac{1}{-36} c^{-15} = -\frac{1}{36} c^{-15} $$ --- 1. **Problem 9:** Simplify $$ \left( \frac{-12m^{-4}n}{7m^{-9}n^{7}} \right)^{0} $$. 2. Any nonzero expression to the zero power is 1. 3. Final expression: $$ 1 $$ --- 1. **Problem 10:** Simplify $ \frac{4d \cdot d^{-8}}{-20 d^{-11}} $. 2. Multiply numerator variables: $ d^{1 + (-8)} = d^{-7} $. 3. Expression becomes $ \frac{4 d^{-7}}{-20 d^{-11}} $. 4. Simplify coefficients: $ \frac{4}{-20} = -\frac{1}{5} $. 5. Simplify $ d $ terms: $ d^{-7 - (-11)} = d^{4} $. 6. Final expression: $$ -\frac{1}{5} d^{4} $$