1. **Problem:** Simplify the expression $$\frac{16m^2}{24m^7}$$. 2. **Formula and rules:** When dividing powers with the same base, subtract the exponents: $$\frac{a^m}{a^n} = a^{m-n}$$. 3. **Simplify coefficients:** $$\frac{16}{24} = \frac{\cancel{8} \times 2}{\cancel{8} \times 3} = \frac{2}{3}$$. 4. **Simplify variables:** $$\frac{m^2}{m^7} = m^{2-7} = m^{-5} = \frac{1}{m^5}$$. 5. **Combine results:** $$\frac{16m^2}{24m^7} = \frac{2}{3} \times \frac{1}{m^5} = \frac{2}{3m^5}$$. 1. **Problem:** Simplify the expression $$\frac{x^2 - 10x - 24}{x + 2}$$. 2. **Formula and rules:** Factor the numerator and then divide by the denominator. 3. **Factor numerator:** Find two numbers that multiply to $$-24$$ and add to $$-10$$: $$-12$$ and $$2$$. 4. **Factorization:** $$x^2 - 10x - 24 = (x - 12)(x + 2)$$. 5. **Divide:** $$\frac{(x - 12)(x + 2)}{x + 2} = x - 12$$ (cancel $$x + 2$$). 1. **Problem:** Simplify the expression $$\frac{4a^2 - 36a}{2a^4 - 24a^3 + 54a^2}$$. 2. **Factor numerator:** $$4a^2 - 36a = 4a(a - 9)$$. 3. **Factor denominator:** Factor out $$2a^2$$ first: $$2a^4 - 24a^3 + 54a^2 = 2a^2(a^2 - 12a + 27)$$. 4. **Factor quadratic:** $$a^2 - 12a + 27$$ factors as $$(a - 9)(a - 3)$$. 5. **Rewrite denominator:** $$2a^2 (a - 9)(a - 3)$$. 6. **Combine:** $$\frac{4a(a - 9)}{2a^2 (a - 9)(a - 3)}$$. 7. **Cancel common factors:** Cancel $$2$$ and $$a - 9$$: $$\frac{\cancel{4}a \cancel{(a - 9)}}{\cancel{2} a^2 \cancel{(a - 9)} (a - 3)} = \frac{2a}{a^2 (a - 3)}$$. 8. **Simplify powers:** $$\frac{2a}{a^2 (a - 3)} = \frac{2}{a (a - 3)}$$. **Final answers:** 1. $$\frac{2}{3m^5}$$ 2. $$x - 12$$ 3. $$\frac{2}{a(a - 3)}$$