The expression provided is:

$\frac{x}{\frac{3}{x} + \frac{1}{x}}$

Let's simplify this step by step:

Step 1: Simplify the denominator

The denominator is $\frac{3}{x} + \frac{1}{x}$. Since both terms have the same denominator, we can add them together:

$\frac{3}{x} + \frac{1}{x} = \frac{3 + 1}{x} = \frac{4}{x}$

Step 2: Simplify the entire expression

Now, substitute the simplified denominator back into the original expression:

$\frac{x}{\frac{4}{x}}$

When dividing by a fraction, we multiply by its reciprocal. So:

$\frac{x}{\frac{4}{x}} = x \cdot \frac{x}{4} = \frac{x^2}{4}$

Final Answer:

The simplified expression is:

$\frac{x^2}{4}$

Would you like more details, or do you have any questions about this process?

Here are 5 related questions you might explore:

1. How can I simplify more complex algebraic fractions?
2. What are the properties of exponents when simplifying expressions?
3. How can the concept of reciprocals help in division of fractions?
4. How do you add fractions with different denominators?
5. How does the structure of a rational function affect its graph?

Tip: Always look for common denominators when adding or subtracting fractions!