Find the third proportional to `(x^(2)-y^(2))` and (x+y)

Answer

To find the third proportional to \( (x^2 - y^2) \) and \( (x + y) \), we can follow these steps:

### Step 1: Set up the proportion
We denote the third proportional as \( z \). The relationship can be expressed as:
\[ \frac{x^2 - y^2}{x + y} = \frac{x + y}{z} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ (x^2 - y^2) \cdot z = (x + y) \cdot (x + y) \]
This simplifies to:
\[ (x^2 - y^2) \cdot z = (x + y)^2 \]

### Step 3: Solve for \( z \)
To isolate \( z \), we divide both sides by \( (x^2 - y^2) \):
\[ z = \frac{(x + y)^2}{(x^2 - y^2)} \]

### Step 4: Factor the denominator
Recall that \( x^2 - y^2 \) can be factored as \( (x - y)(x + y) \):
\[ z = \frac{(x + y)^2}{(x - y)(x + y)} \]

### Step 5: Simplify the expression
We can cancel \( (x + y) \) from the numerator and denominator (assuming \( x + y \neq 0 \)):
\[ z = \frac{x + y}{x - y} \]

### Final Answer
Thus, the third proportional to \( (x^2 - y^2) \) and \( (x + y) \) is:
\[ \frac{x + y}{x - y} \]

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