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

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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:
$$
\text{LCD} = 28x^2 + 4x
$$
The least common denominator (LCD) of $$ \frac{1}{4x} $$ and $$ \frac{3}{7x+1} $$ is $$ 28x^2 + 4x $$.

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

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