1. **Problem:** Simplify $2m^2 \cdot 2m^3$. 2. **Formula:** When multiplying powers with the same base, add the exponents: $a^x \cdot a^y = a^{x+y}$. 3. **Work:** $$2m^2 \cdot 2m^3 = (2 \cdot 2)(m^{2+3}) = 4m^5$$ 4. **Explanation:** Multiply the coefficients $2 \times 2 = 4$ and add exponents of $m$: $2 + 3 = 5$. 1. **Problem:** Simplify $4r^{-3} \cdot 2r^2$. 2. **Formula:** Same as above, add exponents for same base. 3. **Work:** $$4r^{-3} \cdot 2r^2 = (4 \cdot 2) r^{-3+2} = 8r^{-1} = \frac{8}{r}$$ 4. **Explanation:** Multiply coefficients $4 \times 2 = 8$, add exponents $-3 + 2 = -1$, negative exponent means reciprocal. 1. **Problem:** Simplify $4a^3 b^2 \cdot 3a^{-4} b^{-3}$. 2. **Formula:** Multiply coefficients and add exponents for each variable. 3. **Work:** $$4a^3 b^2 \cdot 3a^{-4} b^{-3} = (4 \cdot 3) a^{3 + (-4)} b^{2 + (-3)} = 12 a^{-1} b^{-1} = \frac{12}{ab}$$ 4. **Explanation:** Multiply coefficients $4 \times 3 = 12$, add exponents for $a$: $3 - 4 = -1$, for $b$: $2 - 3 = -1$, negative exponents mean reciprocal. 1. **Problem:** Simplify $(4a^3)^2$. 2. **Formula:** Power of a product: $(ab)^n = a^n b^n$ and power of a power: $(a^m)^n = a^{mn}$. 3. **Work:** $$(4a^3)^2 = 4^2 (a^3)^2 = 16 a^{3 \times 2} = 16 a^6$$ 4. **Explanation:** Square the coefficient $4^2 = 16$, multiply exponents $3 \times 2 = 6$. 1. **Problem:** Simplify $(2x^2)^{-4}$. 2. **Formula:** Negative exponent means reciprocal: $(a^m)^{-n} = \frac{1}{a^{mn}}$. 3. **Work:** $$(2x^2)^{-4} = \frac{1}{(2x^2)^4} = \frac{1}{2^4 x^{2 \times 4}} = \frac{1}{16 x^8}$$ 4. **Explanation:** Negative exponent flips the expression, raise each factor to the 4th power. 1. **Problem:** Simplify $(2x^4 y^{-3})^{-2}$. 2. **Formula:** Same as above for negative exponent. 3. **Work:** $$(2x^4 y^{-3})^{-2} = \frac{1}{(2x^4 y^{-3})^2} = \frac{1}{2^2 x^{4 \times 2} y^{-3 \times 2}} = \frac{1}{4 x^8 y^{-6}} = \frac{1}{4 x^8} y^6 = \frac{y^6}{4 x^8}$$ 4. **Explanation:** Flip due to negative exponent, square each factor, then simplify negative exponent in denominator. 1. **Problem:** Simplify $\frac{12m^4}{3m^2}$. 2. **Formula:** When dividing powers with the same base, subtract exponents: $\frac{a^x}{a^y} = a^{x-y}$. 3. **Work:** $$\frac{12m^4}{3m^2} = \frac{12}{3} m^{4-2} = 4 m^2$$ 4. **Explanation:** Divide coefficients $12 \div 3 = 4$, subtract exponents $4 - 2 = 2$. 1. **Problem:** Simplify $\frac{50x^5 y}{100 x y^3}$. 2. **Formula:** Same as above for division. 3. **Work:** $$\frac{50x^5 y}{100 x y^3} = \frac{50}{100} x^{5-1} y^{1-3} = \frac{1}{2} x^4 y^{-2} = \frac{x^4}{2 y^2}$$ 4. **Explanation:** Simplify fraction $\frac{50}{100} = \frac{1}{2}$, subtract exponents for $x$ and $y$, negative exponent means reciprocal. **Final answers:** 1. $4m^5$ 2. $\frac{8}{r}$ 3. $\frac{12}{ab}$ 4. $16a^6$ 5. $\frac{1}{16 x^8}$ 6. $\frac{y^6}{4 x^8}$ 7. $4 m^2$ 8. $\frac{x^4}{2 y^2}$