To determine the least common denominator (LCD) of the rational expressions [tex]\(\frac{2}{x}\)[/tex] and [tex]\(\frac{3}{x^2}\)[/tex], you should follow these steps:
1. Identify the denominators of each rational expression:
- The denominator of [tex]\(\frac{2}{x}\)[/tex] is [tex]\(x\)[/tex].
- The denominator of [tex]\(\frac{3}{x^2}\)[/tex] is [tex]\(x^2\)[/tex].
2. Factor each denominator into its prime factors:
- For [tex]\(x\)[/tex], the prime factor is [tex]\(x\)[/tex].
- For [tex]\(x^2\)[/tex], the prime factor is [tex]\(x \cdot x\)[/tex], which is [tex]\(x\)[/tex] two times.
3. Determine the LCD by taking the highest power of each distinct prime factor present in any of the denominators:
- The highest power of [tex]\(x\)[/tex] between the denominators [tex]\(x\)[/tex] and [tex]\(x^2\)[/tex] is [tex]\(x^2\)[/tex].
Thus, the least common denominator (LCD) for the expressions [tex]\(\frac{2}{x}\)[/tex] and [tex]\(\frac{3}{x^2}\)[/tex] is [tex]\(\boxed{x^2}\)[/tex].
