1. Stating the problem: Simplify the expression $$\left(\frac{29 - 1}{b - 1}\right)^{-2} : \left(\frac{49 - 2}{b - 3}\right)^{-1}$$. 2. Simplify the numerators inside the parentheses: $$29 - 1 = 28$$ $$49 - 2 = 47$$ So the expression becomes: $$\left(\frac{28}{b - 1}\right)^{-2} : \left(\frac{47}{b - 3}\right)^{-1}$$ 3. Recall that division of expressions is the same as multiplication by the reciprocal: $$A : B = A \times \frac{1}{B}$$ So, $$\left(\frac{28}{b - 1}\right)^{-2} : \left(\frac{47}{b - 3}\right)^{-1} = \left(\frac{28}{b - 1}\right)^{-2} \times \left(\frac{47}{b - 3}\right)^{1}$$ 4. Use the rule for negative exponents: $$x^{-n} = \frac{1}{x^n}$$ So, $$\left(\frac{28}{b - 1}\right)^{-2} = \left(\frac{b - 1}{28}\right)^2 = \frac{(b - 1)^2}{28^2}$$ 5. Substitute back: $$\frac{(b - 1)^2}{28^2} \times \frac{47}{b - 3} = \frac{(b - 1)^2 \times 47}{28^2 \times (b - 3)}$$ 6. Calculate $28^2$: $$28^2 = 784$$ 7. Final simplified expression: $$\frac{47 (b - 1)^2}{784 (b - 3)}$$ This is the simplified form of the original expression.