1. **Stating the problem:** Simplify each expression using index laws. 2. **Recall the index laws:** - $(a^m)^n = a^{m \times n}$ - $a^m \times a^n = a^{m+n}$ - $\frac{a^m}{a^n} = a^{m-n}$ - $\frac{a^m}{a^n} \div a^p = \frac{a^m}{a^{n+p}} = a^{m-(n+p)}$ 3. **Simplify each part:** **a)** $\frac{(3^2)^3}{3^4} = \frac{3^{2 \times 3}}{3^4} = \frac{3^6}{3^4}$ Intermediate step with cancellation: $$\frac{3^{\cancel{4}} \times 3^{2}}{3^{\cancel{4}}} = 3^{6-4} = 3^2$$ Final answer: $3^2$ **b)** $\frac{7^5 \times 7^3}{7^6} = \frac{7^{5+3}}{7^6} = \frac{7^8}{7^6}$ Intermediate step with cancellation: $$\frac{7^{\cancel{6}} \times 7^{2}}{7^{\cancel{6}}} = 7^{8-6} = 7^2$$ Final answer: $7^2$ **c)** $\frac{5^9}{5^{14}} \div 5^7 = \frac{5^9}{5^{14+7}} = \frac{5^9}{5^{21}}$ Intermediate step with cancellation: $$\frac{5^{\cancel{9}}}{5^{12}} = 5^{9-21} = 5^{-12}$$ Final answer: $5^{-12}$ **d)** Simplify $\frac{abc^2 \times a^3 c}{ab^2 \times (c^2)^3}$ First, rewrite powers: $$\frac{a^{1} b^{1} c^{2} \times a^{3} c^{1}}{a^{1} b^{2} c^{2 \times 3}} = \frac{a^{1+3} b^{1} c^{2+1}}{a^{1} b^{2} c^{6}} = \frac{a^{4} b^{1} c^{3}}{a^{1} b^{2} c^{6}}$$ Intermediate step with cancellation: $$\frac{a^{\cancel{1}} a^{3} b^{\cancel{1}} c^{3}}{a^{\cancel{1}} b^{2} c^{6}} = a^{4-1} b^{1-2} c^{3-6} = a^{3} b^{-1} c^{-3}$$ Final answer: $a^{3} b^{-1} c^{-3}$ **Summary:** - a) $3^2$ - b) $7^2$ - c) $5^{-12}$ - d) $a^{3} b^{-1} c^{-3}$