The equation \( a^2 - b^{12} = (a + b)(a - b) \) is not correct. This is because the expression \( a^2 - b^{12} \) cannot be factored as simply \( (a + b)(a - b) \).

Here's why:

1. **Difference of Squares Formula**: The difference of squares formula states that \( x^2 - y^2 = (x + y)(x - y) \). This applies only when the exponents of both terms are 2.

2. **Your Expression**: In \( a^2 - b^{12} \), the exponent on \( b \) is 12, which is much larger than 2. Therefore, you cannot apply the difference of squares formula directly.

### Correct Approach to Factor \( a^2 - b^{12} \):

Instead, you should recognize that \( b^{12} \) is itself a perfect square: \( b^{12} = (b^6)^2 \). Thus, the expression can be factored using the difference of squares formula as follows:

\[
a^2 - b^{12} = a^2 - (b^6)^2 = (a - b^6)(a + b^6)
\]

This is the correct factorization of the given expression.

Would you like more details or have any questions?

Here are some related questions:

1. How can you factor a difference of cubes?
2. What is the sum of squares formula?
3. How would you factor \( a^4 - b^4 \)?
4. Can all polynomials be factored?
5. How do you factor \( a^2 - b^{24} \)?

**Tip:** Always look for patterns like perfect squares or cubes when factoring expressions.

Explore how to factor the expression a^2 - b^12 using the difference of squares formula. Understand the concept of exponents and how to break down complex algebraic expressions. Suitable for students in Grades 9-12.

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