Find
the mean proportional to x - y and ` (x-y)^(3)`

Answer

To find the mean proportional between \( x - y \) and \( (x - y)^3 \), we can follow these steps:

### Step 1: Identify the two numbers
Let \( a = x - y \) and \( b = (x - y)^3 \).

### Step 2: Use the formula for mean proportional
The mean proportional \( m \) between two numbers \( a \) and \( b \) is given by the formula:
\[ m = \sqrt{a \cdot b} \]

### Step 3: Substitute the values of \( a \) and \( b \)
Substituting the values we identified:
\[ m = \sqrt{(x - y) \cdot (x - y)^3} \]

### Step 4: Simplify the expression
We can simplify the expression inside the square root:
\[ m = \sqrt{(x - y)^1 \cdot (x - y)^3} = \sqrt{(x - y)^{1 + 3}} = \sqrt{(x - y)^4} \]

### Step 5: Take the square root
Taking the square root of \( (x - y)^4 \):
\[ m = (x - y)^{4/2} = (x - y)^2 \]

### Final Answer
Thus, the mean proportional to \( x - y \) and \( (x - y)^3 \) is:
\[ (x - y)^2 \]