Sure, let's proceed step-by-step to rewrite the given expressions using a common denominator.
### Expression 1: [tex]\(\frac{2x}{x} + \frac{3}{x^2}\)[/tex]
1. Simplify individual fractions (if possible):
- [tex]\(\frac{2x}{x}\)[/tex] simplifies to [tex]\(2\)[/tex].
- So, the expression becomes [tex]\(2 + \frac{3}{x^2}\)[/tex].
2. Find a common denominator:
- We want both terms to have the same denominator. The least common denominator (LCD) for [tex]\(2\)[/tex] and [tex]\(\frac{3}{x^2}\)[/tex] is [tex]\(x^2\)[/tex].
3. Rewrite each term with the common denominator [tex]\(x^2\)[/tex]:
- [tex]\(2\)[/tex] can be rewritten as [tex]\(\frac{2 \cdot x^2}{x^2}\)[/tex].
- [tex]\(\frac{3}{x^2}\)[/tex] already has the denominator [tex]\(x^2\)[/tex].
4. Combine the fractions:
- [tex]\(\frac{2 \cdot x^2}{x^2} + \frac{3}{x^2} = \frac{2x^2 + 3}{x^2}\)[/tex].
So, [tex]\(\frac{2x}{x}+\frac{3}{x^2}\)[/tex] can be rewritten as [tex]\(\frac{2x^2 + 3}{x^2}\)[/tex].
### Expression 2: [tex]\(\frac{1}{x^2} + \frac{3}{x^2}\)[/tex]
1. Identify the common denominator:
- Both fractions already have the same denominator [tex]\(x^2\)[/tex].
2. Combine the fractions directly:
- [tex]\(\frac{1}{x^2} + \frac{3}{x^2} = \frac{1 + 3}{x^2}\)[/tex].
So, [tex]\(\frac{1}{x^2}+\frac{3}{x^2}\)[/tex] can be rewritten as [tex]\(\frac{1 + 3}{x^2}\)[/tex].
### Expression 3: [tex]\(\frac{2x}{x^2} + \frac{3}{x^2}\)[/tex]
1. Identify the common denominator:
- Both fractions already have the same denominator [tex]\(x^2\)[/tex].
2. Combine the fractions directly:
- [tex]\(\frac{2x}{x^2} + \frac{3}{x^2} = \frac{2x + 3}{x^2}\)[/tex].
So, [tex]\(\frac{2x}{x^2} + \frac{3}{x^2}\)[/tex] can be rewritten as [tex]\(\frac{2x + 3}{x^2}\)[/tex].
To summarize, the three rewritten expressions using the common denominator are:
1. [tex]\(\frac{2x}{x} + \frac{3}{x^2}\)[/tex] becomes [tex]\(\frac{2x^2 + 3}{x^2}\)[/tex].
2. [tex]\(\frac{1}{x^2} + \frac{3}{x^2}\)[/tex] becomes [tex]\(\frac{1 + 3}{x^2}\)[/tex].
3. [tex]\(\frac{2x}{x^2} + \frac{3}{x^2}\)[/tex] becomes [tex]\(\frac{2x + 3}{x^2}\)[/tex].
