Question
20. \( \frac{m-2}{3 m+9} \cdot \frac{2 m+6}{2 m-4} \)
Ask by Patel Nguyen. in the United States
Jan 22,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified expression is \( \frac{1}{3} \).
Solution
To simplify the expression:
\[
\frac{m - 2}{3m + 9} \cdot \frac{2m + 6}{2m - 4}
\]
**Step 1: Factor where possible**
1. **Factor the denominators and numerators:**
- \(3m + 9 = 3(m + 3)\)
- \(2m + 6 = 2(m + 3)\)
- \(2m - 4 = 2(m - 2)\)
2. **Rewrite the expression with these factors:**
\[
\frac{m - 2}{3(m + 3)} \cdot \frac{2(m + 3)}{2(m - 2)}
\]
**Step 2: Cancel common factors**
1. **Cancel \(m - 2\) in the numerator and denominator:**
\[
\frac{\cancel{m - 2}}{3(m + 3)} \cdot \frac{2(m + 3)}{2\cancel{(m - 2)}}
\]
2. **Cancel \(m + 3\) in the numerator and denominator:**
\[
\frac{1}{3\cancel{(m + 3)}} \cdot \frac{2\cancel{(m + 3)}}{2}
\]
3. **Cancel the 2's:**
\[
\frac{1}{3} \cdot \frac{\cancel{2}}{\cancel{2}} = \frac{1}{3}
\]
**Final Simplified Expression:**
\[
\frac{1}{3}
\]
**Note:**
- The expression is valid as long as the original denominators are not zero. Therefore, \(m \neq -3\) and \(m \neq 2\).
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Simplify this solution The Deep Dive
To simplify the expression \( \frac{m-2}{3 m+9} \cdot \frac{2 m+6}{2 m-4} \), we start by factoring any common terms. First, factor each part: 1. \( 3m + 9 = 3(m + 3) \) 2. \( 2m + 6 = 2(m + 3) \) 3. \( 2m - 4 = 2(m - 2) \) Now, substitute the factored forms into the expression: \[ \frac{m-2}{3(m+3)} \cdot \frac{2(m+3)}{2(m-2)} \] Next, we can cancel out the common factors from the numerator and the denominator: - The \( m + 3 \) terms cancel out. - The \( m - 2 \) terms also cancel out. This leaves us with: \[ \frac{2}{3} \] Thus, the simplified form of the expression is: \[ \boxed{\frac{2}{3}} \]
