https://r2.erweima.ai/mathbot/image/1723328594855rx1obywx.jpg

The expression in the image is \( x^{4a} - 1 \). To factorize this, we can recognize it as a difference of squares.

### Step-by-Step Factorization

1. **Recognize the difference of squares:**
   \[
   x^{4a} - 1 = (x^{2a})^2 - 1^2
   \]
   This can be written as:
   \[
   x^{4a} - 1 = (x^{2a} - 1)(x^{2a} + 1)
   \]

2. **Further factorization of \(x^{2a} - 1\):**
   Notice that \(x^{2a} - 1\) is also a difference of squares:
   \[
   x^{2a} - 1 = (x^a - 1)(x^a + 1)
   \]

3. **Final factorization:**
   The fully factorized form is:
   \[
   x^{4a} - 1 = (x^a - 1)(x^a + 1)(x^{2a} + 1)
   \]

So, the factorized form of \( x^{4a} - 1 \) is \( (x^a - 1)(x^a + 1)(x^{2a} + 1) \).

Would you like more details on this factorization or have any other questions?

Here are some related questions:

1. How do you factor the difference of squares in general?
2. What are the conditions under which \( x^{4a} - 1 \) can be further factored?
3. Can you explain the difference between factoring and simplifying?
4. How do you factor trinomials like \( x^2 + bx + c \)?
5. How do you recognize and factor perfect square trinomials?
6. How would you factor \( x^{4a} + 1 \), and why is it different from \( x^{4a} - 1 \)?
7. What is the significance of the term "difference of squares"?
8. How do you solve equations involving expressions like \( x^{4a} - 1 \)?

**Tip:** Always check if a polynomial can be further factored, especially when dealing with expressions like differences of squares, sums of cubes, or differences of cubes.

Understand how to factorize the expression x^(4a) - 1 step-by-step. Learn the key concept of difference of squares and its application in algebra. Suitable for high school students and anyone studying advanced algebraic factorization.

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