Simplify: `(x^2)^(n-1) + (y^2)^(n-1) (x^2)^(n-1) - (y^2)^(n-1)`

Options

A. `(x^2)^n - (y^2)^n`
B. `x^2n - y^2n`
C. `(x^n - y^n)+(x^ny^n)`
D. `(x^2 - y^2)^(n-1)`

Answer

To simplify the expression \((x^2)^{n-1} + (y^2)^{n-1} \cdot (x^2)^{n-1} - (y^2)^{n-1}\), we can follow these steps:

### Step 1: Apply the Power of a Power Property
Using the property \((a^m)^n = a^{m \cdot n}\), we can simplify each term in the expression:

\[ (x^2)^{n-1} = x^{2(n-1)} \quad \text{and} \quad (y^2)^{n-1} = y^{2(n-1)} \]

So, we rewrite the expression:

\[ x^{2(n-1)} + y^{2(n-1)} \cdot x^{2(n-1)} - y^{2(n-1)} \]

### Step 2: Combine Like Terms
Now, we can factor out \(y^{2(n-1)}\) from the second term:

\[ x^{2(n-1)} + y^{2(n-1)} \cdot x^{2(n-1)} - y^{2(n-1)} = x^{2(n-1)} + y^{2(n-1)}(x^{2(n-1)} - 1) \]

### Step 3: Factor the Expression
Notice that \(x^{2(n-1)} - 1\) can be factored as a difference of squares if \(n-1\) is greater than or equal to 1:

\[ x^{2(n-1)} - 1 = (x^{n-1} - 1)(x^{n-1} + 1) \]

Thus, we rewrite the expression:

\[ x^{2(n-1)} + y^{2(n-1)}((x^{n-1} - 1)(x^{n-1} + 1)) \]

### Final Expression
The final simplified expression is:

\[ x^{2(n-1)} + y^{2(n-1)}(x^{n-1} - 1)(x^{n-1} + 1) \]