1. Problem: Simplify the quotient $\frac{a^8}{a^3}$. Use the rule for dividing powers with the same base: $$\frac{x^m}{x^n} = x^{m-n}$$ Step 1: Apply the rule: $$\frac{a^8}{a^3} = a^{8-3}$$ Step 2: Simplify the exponent: $$a^5$$ Answer: $a^5$ 2. Problem: Simplify the quotient $\frac{7^{11}}{7^8}$. Step 1: Apply the rule: $$\frac{7^{11}}{7^8} = 7^{11-8}$$ Step 2: Simplify the exponent: $$7^3$$ Answer: $7^3$ 3. Problem: Simplify the quotient $\frac{7 \cdot b^3}{b^4}$. Step 1: Separate constants and variables: $$7 \cdot \frac{b^3}{b^4}$$ Step 2: Apply the rule for powers: $$7 \cdot b^{3-4} = 7 \cdot b^{-1}$$ Step 3: Rewrite negative exponent: $$7 \cdot \frac{1}{b} = \frac{7}{b}$$ Answer: $\frac{7}{b}$ 4. Problem: Simplify the quotient $\frac{x^{10}}{x^7}$. Step 1: Apply the rule: $$x^{10-7} = x^3$$ Answer: $x^3$ 5. Problem: Simplify the quotient $\frac{12 \cdot g^8 \cdot h^4}{g^4 \cdot h^3}$. Step 1: Separate constants and variables: $$12 \cdot \frac{g^8}{g^4} \cdot \frac{h^4}{h^3}$$ Step 2: Apply the rule for powers: $$12 \cdot g^{8-4} \cdot h^{4-3} = 12 \cdot g^4 \cdot h^1$$ Step 3: Simplify: $$12 g^4 h$$ Answer: $12 g^4 h$ 6. Problem: Simplify the quotient $\frac{4 \cdot p^{11}}{8 \cdot p^6}$. Step 1: Simplify constants: $$\frac{4}{8} = \frac{\cancel{4}^1}{\cancel{8}^2} = \frac{1}{2}$$ Step 2: Apply the rule for powers: $$p^{11-6} = p^5$$ Step 3: Combine: $$\frac{1}{2} p^5 = \frac{p^5}{2}$$ Answer: $\frac{p^5}{2}$ 7. Problem: Simplify the quotient $\frac{c^6}{6 c^4}$. Step 1: Separate constants and variables: $$\frac{c^6}{6 c^4} = \frac{c^6}{6 \cdot c^4}$$ Step 2: Apply the rule for powers: $$\frac{c^{6-4}}{6} = \frac{c^2}{6}$$ Answer: $\frac{c^2}{6}$ 8. Problem: Simplify the quotient $\frac{2 \cdot x^3 \cdot y^8}{4 \cdot y^2}$. Step 1: Simplify constants: $$\frac{2}{4} = \frac{\cancel{2}^1}{\cancel{4}^2} = \frac{1}{2}$$ Step 2: Apply the rule for powers on $y$: $$y^{8-2} = y^6$$ Step 3: Combine: $$\frac{1}{2} x^3 y^6 = \frac{x^3 y^6}{2}$$ Answer: $\frac{x^3 y^6}{2}$ 9. Problem: Simplify the quotient $\frac{3 \cdot x^{10} \cdot y^{11}}{18 x^2}$. Step 1: Simplify constants: $$\frac{3}{18} = \frac{\cancel{3}^1}{\cancel{18}^6} = \frac{1}{6}$$ Step 2: Apply the rule for powers on $x$: $$x^{10-2} = x^8$$ Step 3: Combine: $$\frac{1}{6} x^8 y^{11} = \frac{x^8 y^{11}}{6}$$ Answer: $\frac{x^8 y^{11}}{6}$