Find the compounded ratio of:
`(x+y): (x-y), (x^(2) + y^(2)): (x+y)^(2) and (x^(2)-y^(2))^(2): (x^(4)-y^(4))`

Answer

To find the compounded ratio of the given expressions, we will follow these steps:

1. **Identify the Ratios**: We have three ratios to consider:
- \( (x+y) : (x-y) \)
- \( (x^2 + y^2) : (x+y)^2 \)
- \( (x^2 - y^2)^2 : (x^4 - y^4) \)

2. **Write the Compound Ratio Formula**: The compound ratio of \( a : b, c : d, e : f \) is given by:
\[ \text{Compound Ratio} = \frac{a \cdot c \cdot e}{b \cdot d \cdot f} \]

3. **Apply the Formula**: Using the above formula, we can write:
\[ \text{Compound Ratio} = \frac{(x+y) \cdot (x^2 + y^2) \cdot (x^2 - y^2)^2}{(x-y) \cdot (x+y)^2 \cdot (x^4 - y^4)} \]

4. **Simplify Each Term**:
- The term \( (x^4 - y^4) \) can be factored as \( (x^2 - y^2)(x^2 + y^2) \).
- The term \( (x^2 - y^2) \) can be factored as \( (x-y)(x+y) \).

5. **Substituting the Factored Forms**:
\[ \text{Compound Ratio} = \frac{(x+y) \cdot (x^2 + y^2) \cdot (x^2 - y^2)^2}{(x-y) \cdot (x+y)^2 \cdot (x^2 - y^2)(x^2 + y^2)} \]

6. **Cancel Common Terms**:
- The \( (x+y) \) in the numerator and one \( (x+y) \) in the denominator cancel out.
- The \( (x^2 + y^2) \) in the numerator and denominator also cancel out.
- The \( (x^2 - y^2) \) can be expressed as \( (x-y)(x+y) \), and we can cancel one \( (x-y) \) from the numerator.

7. **Final Simplification**:
After canceling, we have:
\[ \text{Compound Ratio} = \frac{(x-y)^2}{(x-y)(x+y)} = \frac{x-y}{x+y} \]

Thus, the compounded ratio is:
\[ \frac{x-y}{x+y} \]