least common denominator of \( \frac{6 x}{x+5} \) and \( \frac{1}{5 x} \)

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 \( 5x(x+5) \).