Find the compounded ratio of:
`(a-b): (a+b) and (b^(2) + ab): (a^(2) -ab)`

Answer

To find the compounded ratio of \((a-b):(a+b)\) and \((b^2 + ab):(a^2 - ab)\), we will follow these steps:

### Step 1: Write the compounded ratio
The compounded ratio of two ratios \((x:y)\) and \((z:w)\) is given by:
\[ \frac{x \cdot z}{y \cdot w} \]
In our case, we have:
\[ x = (a-b), \quad y = (a+b), \quad z = (b^2 + ab), \quad w = (a^2 - ab) \]
Thus, the compounded ratio can be expressed as:
\[ \frac{(a-b)(b^2 + ab)}{(a+b)(a^2 - ab)} \]

### Step 2: Expand the numerator and denominator
Now we will expand both the numerator and the denominator.

**Numerator:**
\[ (a-b)(b^2 + ab) = ab^2 + a^2b - b^3 - ab^2 = a^2b - b^3 \]

**Denominator:**
\[ (a+b)(a^2 - ab) = a^3 - a^2b + ab^2 - b^3 \]

### Step 3: Simplify the expression
Now we can simplify the expression:
\[ \frac{a^2b - b^3}{a^3 - a^2b + ab^2 - b^3} \]

### Step 4: Factor out common terms
We can factor out \(b\) from the numerator:
\[ = \frac{b(a^2 - b^2)}{(a^3 - b^3)} \]
Using the difference of cubes, we can rewrite the denominator:
\[ = \frac{b(a^2 - b^2)}{(a-b)(a^2 + ab + b^2)} \]

### Step 5: Cancel common factors
Now we can cancel out \(a-b\) from the numerator and denominator:
\[ = \frac{b(a+b)}{(a^2 + ab + b^2)} \]

### Step 6: Final ratio
Thus, the compounded ratio simplifies to:
\[ \frac{b(a+b)}{(a^2 + ab + b^2)} \]

### Conclusion
The final compounded ratio is:
\[ b : (a^2 + ab + b^2) \]