Let's go through the simplification step by step.

### Given Expression:

\[
\frac{1}{x+y} + \frac{x+y}{x^3 + y^3}
\]

### Step 1: Find a Common Denominator

The two fractions have different denominators: \( x+y \) and \( x^3 + y^3 \). To combine them, we need a common denominator, which is the product of these two denominators:

\[
\text{Common denominator} = (x+y)(x^3 + y^3)
\]

### Step 2: Rewrite Each Fraction with the Common Denominator

- For the first fraction, \(\frac{1}{x+y}\), multiply both the numerator and the denominator by \( x^3 + y^3 \):

\[
\frac{1 \cdot (x^3 + y^3)}{(x+y)(x^3 + y^3)} = \frac{x^3 + y^3}{(x+y)(x^3 + y^3)}
\]

- For the second fraction, \(\frac{x+y}{x^3 + y^3}\), multiply both the numerator and the denominator by \( x+y \):

\[
\frac{(x+y) \cdot (x+y)}{(x+y)(x^3 + y^3)} = \frac{(x+y)^2}{(x+y)(x^3 + y^3)}
\]

### Step 3: Combine the Fractions

Now that both fractions have the same denominator, we can add them:

\[
\frac{x^3 + y^3}{(x+y)(x^3 + y^3)} + \frac{(x+y)^2}{(x+y)(x^3 + y^3)} = \frac{x^3 + y^3 + (x+y)^2}{(x+y)(x^3 + y^3)}
\]

### Step 4: Simplify the Resulting Expression

The expression in the numerator is:

\[
x^3 + y^3 + (x+y)^2
\]

So the final simplified expression is:

\[
\frac{x^3 + y^3 + (x+y)^2}{(x+y)(x^3 + y^3)}
\]

This is the simplified form of the original expression.

Would you like to explore any further details or have any questions?

Here are five related questions you might consider:
1. How can you factor \( (x+y)^2 \) and \( x^3 + y^3 \) further?
2. What special cases arise when \( x \) or \( y \) is zero?
3. How does the expression behave as \( x \) and \( y \) approach infinity?
4. What if the original expression is integrated with respect to \( x \)?
5. Can you evaluate the expression for specific values of \( x \) and \( y \)?

**Tip:** When simplifying algebraic expressions, always look for common factors in the numerators and denominators to reduce complexity.

Learn how to simplify the fractional expression 1/(x+y) + (x+y)/(x^3 + y^3) step-by-step. This problem involves finding a common denominator, combining fractions, and simplifying algebraic expressions. Suitable for high school students.

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