The third proportional of the following numbers ` ( x - y)^(2) , ( x^(2) - y^(2))^(2)` is

Options

A. `( x + y)^(3) ( x - y) ^(2)`
B. `(x + y)^(4) ( x - y) ^(2)`
C. `( x + y )^(2) ( x - y)^(2)`
D. `( x + y )^(2) ( x - y)^(3)`

Answer

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

### Step 1: Understand the concept of third proportional
The third proportional \( z \) to two numbers \( a \) and \( b \) is defined such that the ratio \( \frac{a}{b} = \frac{b}{z} \). This can be rearranged to find \( z \) as:
\[ z = \frac{b^2}{a} \]

### Step 2: Identify the given numbers
In our case, let:
- \( a = (x - y)^2 \)
- \( b = (x^2 - y^2)^2 \)

### Step 3: Calculate \( b^2 \)
First, we need to compute \( b^2 \):
\[ b^2 = ((x^2 - y^2)^2)^2 = (x^2 - y^2)^4 \]

### Step 4: Calculate \( z \)
Now, we substitute \( a \) and \( b^2 \) into the formula for the third proportional:
\[ z = \frac{b^2}{a} = \frac{(x^2 - y^2)^4}{(x - y)^2} \]

### Step 5: Simplify \( z \)
Next, we can simplify \( z \). We know that:
\[ x^2 - y^2 = (x + y)(x - y) \]
Thus,
\[ (x^2 - y^2)^4 = ((x + y)(x - y))^4 = (x + y)^4 (x - y)^4 \]
Now substituting this back into our equation for \( z \):
\[ z = \frac{(x + y)^4 (x - y)^4}{(x - y)^2} \]
This simplifies to:
\[ z = (x + y)^4 (x - y)^{4 - 2} = (x + y)^4 (x - y)^2 \]

### Final Answer
Thus, the third proportional of the numbers \( (x - y)^2 \) and \( (x^2 - y^2)^2 \) is:
\[ z = (x + y)^4 (x - y)^2 \]

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