east common denominator of \( \frac{1}{4 x} \) and \( \frac{3}{7 x+1} \)

To find the least common denominator (LCD) of the fractions \( \frac{1}{4x} \) and \( \frac{3}{7x+1} \), we need to identify the denominators of both fractions: 1. The first fraction has a denominator of \( 4x \). 2. The second fraction has a denominator of \( 7x + 1 \). The least common denominator is the least common multiple (LCM) of these two denominators. ### Step 1: Factor the denominators - The first denominator \( 4x \) can be factored as \( 4 \cdot x \). - The second denominator \( 7x + 1 \) is already in its simplest form. ### Step 2: Determine the LCM The LCM will include each factor at its highest power: - From \( 4x \), we take \( 4 \) and \( x \). - From \( 7x + 1 \), we take \( 7x + 1 \). Thus, the least common denominator (LCD) is: \[ \text{LCD} = 4x(7x + 1) \] Now, let's calculate the LCD explicitly. ### Step 3: Calculate the LCD \[ \text{LCD} = 4x(7x + 1) = 4x \cdot 7x + 4x \cdot 1 = 28x^2 + 4x \] Therefore, the least common denominator of \( \frac{1}{4x} \) and \( \frac{3}{7x+1} \) is: \[ \text{LCD} = 28x^2 + 4x \]