1. Simplify $9x^{-6}y^{3} \div 0.5x^{-3}y^{1}$. Step 1: Rewrite division as multiplication: $$9x^{-6}y^{3} \times \frac{1}{0.5x^{-3}y^{1}}$$ Step 2: Simplify $\frac{1}{0.5} = 2$. Step 3: Combine factors: $$9 \times 2 \times x^{-6} \times x^{3} \times y^{3} \times y^{-1} = 18x^{-3}y^{2}$$ Final answer: $18x^{-3}y^{2}$. 2. Simplify $\left(\frac{125a^{21}}{b^{6}}\right)^{3} \div \left(\frac{b^{3}}{125a^{21}}\right)^{2}$. Step 1: Expand powers: $$\frac{(125)^{3}a^{63}}{b^{18}} \div \frac{b^{6}}{(125)^{2}a^{42}}$$ Step 2: Division of fractions equals multiplication by reciprocal: $$\frac{(125)^{3}a^{63}}{b^{18}} \times \frac{(125)^{2}a^{42}}{b^{6}}$$ Step 3: Multiply numerators and denominators: $$\frac{(125)^{5}a^{105}}{b^{24}}$$ Final answer: $\frac{(125)^{5}a^{105}}{b^{24}}$. 3. Simplify $\sqrt{25}x^{12} \div (5y^{4})^{2} \div 3$. Step 1: Simplify square root: $$\sqrt{25} = 5$$ Step 2: Expand denominator: $$(5y^{4})^{2} = 5^{2}y^{8} = 25y^{8}$$ Step 3: Substitute back: $$\frac{5x^{12}}{25y^{8}} \div 3 = \frac{5x^{12}}{25y^{8}} \times \frac{1}{3} = \frac{5x^{12}}{75y^{8}}$$ Step 4: Simplify: $$\frac{1}{15} \frac{x^{12}}{y^{8}}$$ Final answer: $\frac{x^{12}}{15y^{8}}$. 4. Simplify $7x^{-3}(x^{2})^{13} \div (x^{-2})^{5}$. Step 1: Simplify powers: $$(x^{2})^{13} = x^{26}$$ $$(x^{-2})^{5} = x^{-10}$$ Step 2: Substitute: $$7x^{-3}x^{26} \div x^{-10} = 7x^{23} \div x^{-10}$$ Step 3: Division means subtract exponents: $$7x^{23 - (-10)} = 7x^{33}$$ Final answer: $7x^{33}$. 5. Simplify $5x^{2}(x^{2})^{-9} \div x^{5}$. Step 1: Simplify power: $$(x^{2})^{-9} = x^{-18}$$ Step 2: Substitute: $$5x^{2}x^{-18} \div x^{5} = 5x^{-16} \div x^{5}$$ Step 3: Division subtracts exponents: $$5x^{-16 - 5} = 5x^{-21}$$ Final answer: $5x^{-21}$. 6. Simplify $\frac{14(a-1)}{(35(a-1))^{2}}$. Step 1: Expand denominator: $$(35(a-1))^{2} = 35^{2}(a-1)^{2} = 1225(a-1)^{2}$$ Step 2: Substitute: $$\frac{14(a-1)}{1225(a-1)^{2}}$$ Step 3: Simplify coefficients: $$\frac{14}{1225} = \frac{2}{175}$$ Step 4: Simplify powers of $(a-1)$: $$(a-1)^{1} \div (a-1)^{2} = (a-1)^{-1} = \frac{1}{a-1}$$ Final answer: $\frac{2}{175(a-1)}$. Total questions solved: 6.