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

Answer

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

### Step 1: Identify the expression
We have the expression in the form of \((a + b)(a - b)\), where:
- \(a = x^{2^{(n-1)}}\)
- \(b = y^{2^{(n-1)}}\)

### Step 2: Apply the difference of squares formula
Using the difference of squares formula, which states that \((a + b)(a - b) = a^2 - b^2\), we can rewrite our expression:
\[ (x^{2^{(n-1)}} + y^{2^{(n-1)}})(x^{2^{(n-1)}} - y^{2^{(n-1)}}) = a^2 - b^2 \]

### Step 3: Calculate \(a^2\) and \(b^2\)
Now, we calculate \(a^2\) and \(b^2\):
- \(a^2 = (x^{2^{(n-1)}})^2 = x^{2 \cdot 2^{(n-1)}} = x^{2^{n}}\)
- \(b^2 = (y^{2^{(n-1)}})^2 = y^{2 \cdot 2^{(n-1)}} = y^{2^{n}}\)

### Step 4: Substitute back into the difference of squares formula
Now we substitute \(a^2\) and \(b^2\) back into the equation:
\[ x^{2^{n}} - y^{2^{n}} \]

### Final Result
Thus, the simplified expression is:
\[ x^{2^{n}} - y^{2^{n}} \]

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