Factorisation of `a^(2x)-b^(2x)` is

Options

A. `(a^(x)+b^(x))(a^(x)-b^(x))`
B. `(a^(x)-b^(x))^(2)`
C. `(a^(x)+b^(x))(a^(2)-b^(2))`
D. `(a^(x)-b^(x))(a^(2)+b^(2))`

Answer

To factorize the expression \( a^{2x} - b^{2x} \), we can follow these steps:

### Step 1: Recognize the expression
The expression \( a^{2x} - b^{2x} \) is a difference of squares. We can rewrite it as:
\[ a^{2x} - b^{2x} = (a^x)^2 - (b^x)^2 \]

### Step 2: Apply the difference of squares formula
We know from algebra that the difference of squares can be factored using the identity:
\[ A^2 - B^2 = (A + B)(A - B) \]
In our case, let \( A = a^x \) and \( B = b^x \). Therefore, we can apply the formula:
\[ (a^x)^2 - (b^x)^2 = (a^x + b^x)(a^x - b^x) \]

### Step 3: Write the final factorized form
Thus, the factorization of \( a^{2x} - b^{2x} \) is:
\[ a^{2x} - b^{2x} = (a^x + b^x)(a^x - b^x) \]

### Final Answer:
The factorization of \( a^{2x} - b^{2x} \) is:
\[ (a^x + b^x)(a^x - b^x) \]

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