1. **Problem Statement:** Find the LCM (Least Common Multiple) and HCF (Highest Common Factor) of the algebraic expressions given in part (a): $45xy$ and $135x^2y$. 2. **Formulas and Rules:** - The HCF of algebraic expressions is found by taking the product of the lowest powers of all common factors. - The LCM is found by taking the product of the highest powers of all factors present in either expression. 3. **Step-by-step Solution for (a):** - Expressions: $45xy$ and $135x^2y$ - Prime factorize the coefficients: $$45 = 3^2 \times 5$$ $$135 = 3^3 \times 5$$ - Variables: - First expression has $x^1 y^1$ - Second expression has $x^2 y^1$ 4. **Find HCF:** - Coefficients: minimum powers of primes common to both: $$3^{\min(2,3)} = 3^2 = 9$$ $$5^{\min(1,1)} = 5$$ - Variables: minimum powers common: $$x^{\min(1,2)} = x^1 = x$$ $$y^{\min(1,1)} = y$$ - So, $$\text{HCF} = 9 \times 5 \times x \times y = 45xy$$ 5. **Find LCM:** - Coefficients: maximum powers of primes: $$3^{\max(2,3)} = 3^3 = 27$$ $$5^{\max(1,1)} = 5$$ - Variables: maximum powers: $$x^{\max(1,2)} = x^2$$ $$y^{\max(1,1)} = y$$ - So, $$\text{LCM} = 27 \times 5 \times x^2 \times y = 135x^2y$$ 6. **Final answers:** - HCF = $45xy$ - LCM = $135x^2y$