Find the LCD of each pair of expressions. \( \begin{array}{ll}\text { 17. } \frac{1}{2} ; \frac{4}{x^{2}} & \text { 18. } \frac{b}{6} ; \frac{2 b}{9} \\ \text { 21. } \frac{3}{5} ; \frac{x}{x+2} & \text { 22. } \frac{2}{a b} ; \frac{a-b}{b^{2} c}\end{array} \)

To find the Least Common Denominator (LCD) for each pair of fractions, we'll identify the least common multiple (LCM) of the denominators in each pair. Here's the step-by-step process for each problem: ### **Problem 17: \(\frac{1}{2}\) and \(\frac{4}{x^{2}}\)** - **Denominators:** 2 and \(x^{2}\) - **LCD:** The LCD is the product of the distinct factors, which are 2 and \(x^{2}\). \[ \text{LCD} = 2x^{2} \] ### **Problem 18: \(\frac{b}{6}\) and \(\frac{2b}{9}\)** - **Denominators:** 6 and 9 - **Prime Factorization:** - 6 = \(2 \times 3\) - 9 = \(3^{2}\) - **LCD:** The LCM of 6 and 9 is \(2 \times 3^{2} = 18\). \[ \text{LCD} = 18 \] ### **Problem 21: \(\frac{3}{5}\) and \(\frac{x}{x+2}\)** - **Denominators:** 5 and \(x + 2\) - **LCD:** Since 5 and \(x + 2\) are distinct and have no common factors, the LCD is their product. \[ \text{LCD} = 5(x + 2) \] ### **Problem 22: \(\frac{2}{ab}\) and \(\frac{a - b}{b^{2}c}\)** - **Denominators:** \(ab\) and \(b^{2}c\) - **Factorization:** - \(ab = a \times b\) - \(b^{2}c = b \times b \times c\) - **LCD:** Take the highest power of each distinct factor. \[ \text{LCD} = a \times b^{2} \times c = ab^{2}c \] ### **Summary of LCDs:** 1. **Problem 17:** \(2x^{2}\) 2. **Problem 18:** \(18\) 3. **Problem 21:** \(5(x + 2)\) 4. **Problem 22:** \(ab^{2}c\) Answer: Problem 17 Answer: The LCD is \(2\,x^{2}\). Problem 18 Answer: The LCD is \(18\). Problem 21 Answer: The LCD is \(5(x + 2)\). Problem 22 Answer: The LCD is \(a\,b^{2}c\).