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

Question

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

UpStudy AI · Accepted Answer

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 $$a\,b^{2}c$$.
The LCDs are: - Problem 17: $$2x^{2}$$ - Problem 18: $$18$$ - Problem 21: $$5(x + 2)$$ - Problem 22: $$ab^{2}c$$

Find the LCD of each pair of expressions.

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