Question
least common denominator of \( \frac{6 x}{x+5} \) and \( \frac{1}{5 x} \)
Ask by Weaver Peters. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The least common denominator is \( 5x(x+5) \).
Solution
1. Write down the denominators of the two fractions:
- For \( \frac{6x}{x+5} \), the denominator is \( x+5 \).
- For \( \frac{1}{5x} \), the denominator is \( 5x \).
2. Factor each denominator into its prime factors:
- \( x+5 \) is already a prime factor.
- \( 5x \) factors as \( 5 \) and \( x \).
3. The least common denominator (LCD) must include each unique factor at its highest power:
- The factor \( 5 \) appears in \( 5x \).
- The factor \( x \) appears in \( 5x \).
- The factor \( x+5 \) appears in the first denominator.
4. Multiply the unique factors together to get the LCD:
\[
\text{LCD} = 5x(x+5)
\]
Thus, the least common denominator is \( 5x(x+5) \).
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Simplify this solution Bonus Knowledge
To find the least common denominator (LCD) of the fractions \( \frac{6 x}{x+5} \) and \( \frac{1}{5 x} \), we first identify the denominators: \( x + 5 \) and \( 5x \). The LCD is obtained by taking the highest power of all unique factors present in both denominators. The factors of the first denominator \( x + 5 \) are simply \( x + 5 \), and the factors of the second denominator \( 5x \) are \( 5 \) and \( x \). Therefore, the least common denominator is obtained by combining these factors: \[ \text{LCD} = 5(x + 5) = 5x + 25. \] Thus, the least common denominator of the two fractions is \( 5(x + 5) \).
