(a-2b)^3 answer first then soulution, then what method should I use? FOIL or box?

### Answer:
The expression \((a - 2b)^3\) expands to \(a^3 - 6a^2b + 12ab^2 - 8b^3\).

### Solution:

To expand \((a - 2b)^3\), we can follow these steps:

1. **Start with the Binomial Theorem**: The Binomial Theorem states that \((x + y)^n\) can be expanded as:
   \[
   \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
   \]
   Here, \(x = a\), \(y = -2b\), and \(n = 3\).

2. **Apply the Binomial Expansion**: Expand \((a - 2b)^3\) using the binomial coefficients:
   \[
   (a - 2b)^3 = \binom{3}{0}a^3(-2b)^0 + \binom{3}{1}a^2(-2b)^1 + \binom{3}{2}a^1(-2b)^2 + \binom{3}{3}a^0(-2b)^3
   \]
   Simplifying each term:
   - \( \binom{3}{0}a^3(-2b)^0 = 1 \cdot a^3 \cdot 1 = a^3 \)
   - \( \binom{3}{1}a^2(-2b)^1 = 3 \cdot a^2 \cdot (-2b) = -6a^2b \)
   - \( \binom{3}{2}a(-2b)^2 = 3 \cdot a \cdot 4b^2 = 12ab^2 \)
   - \( \binom{3}{3}(-2b)^3 = 1 \cdot (-8b^3) = -8b^3 \)

3. **Combine the terms**: Putting it all together gives:
   \[
   (a - 2b)^3 = a^3 - 6a^2b + 12ab^2 - 8b^3
   \]

### Method: FOIL or Box?

For this specific problem:

- **Neither FOIL nor the box method** is typically used directly to expand a binomial raised to a power greater than 2. 
- **Binomial expansion** is more appropriate. 

However, if you are expanding a cubic binomial like this by multiplying step-by-step, then **Box method** (also known as area method) can be useful for organizing the multiplication in stages. For example, first multiply \((a - 2b)\) by \((a - 2b)\), then multiply the result by \((a - 2b)\) again.

Would you like more details on any of these steps or have any questions?

### Follow-Up Questions:
1. How would you expand \((a + 3b)^3\) using the Binomial Theorem?
2. Can you use the Box method to expand \((a - 2b)^2\)?
3. What are the advantages of using the Binomial Theorem over FOIL for higher powers?
4. How does the sign of the middle terms change when expanding \((a + b)^3\) compared to \((a - b)^3\)?
5. What is the difference between the Binomial Theorem and Pascal's Triangle in expansion?

### Tip:
When expanding binomials, remember that the Binomial Theorem works efficiently for any power, especially when the exponent is greater than 2.

Learn how to expand (a-2b)^3 using the Binomial Theorem with detailed steps and solution. Understand the application of binomial coefficients and how to simplify each term effectively. Suitable for high school students and anyone interested in algebraic expansions.

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