Use Pascal's triangle to expand the expression (3a + 2b)⁷?

To expand \((3a + 2b)^7\) using Pascal's Triangle, we follow these steps:

1. **Identify the coefficients using Pascal's Triangle for \(n = 7\):**
   The 7th row of Pascal's Triangle is:
   \[
   1, 7, 21, 35, 35, 21, 7, 1
   \]

2. **Apply the binomial theorem:**
   The binomial expansion for \((x + y)^n\) is given by:
   \[
   (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
   \]
   For \((3a + 2b)^7\), \(x = 3a\), \(y = 2b\), and \(n = 7\). Now, expand term by term:

\[
(3a + 2b)^7 = \sum_{k=0}^{7} \binom{7}{k} (3a)^{7-k} (2b)^k
\]

Using the coefficients from Pascal's Triangle, we get:

\[
= 1(3a)^7(2b)^0 + 7(3a)^6(2b)^1 + 21(3a)^5(2b)^2 + 35(3a)^4(2b)^3 + 35(3a)^3(2b)^4 + 21(3a)^2(2b)^5 + 7(3a)^1(2b)^6 + 1(3a)^0(2b)^7
\]

3. **Simplify each term:**

- First term: \((3a)^7 = 2187a^7\)
- Second term: \(7(3a)^6(2b) = 7 \cdot 729a^6 \cdot 2b = 10206a^6b\)
- Third term: \(21(3a)^5(2b)^2 = 21 \cdot 243a^5 \cdot 4b^2 = 20412a^5b^2\)
- Fourth term: \(35(3a)^4(2b)^3 = 35 \cdot 81a^4 \cdot 8b^3 = 22680a^4b^3\)
- Fifth term: \(35(3a)^3(2b)^4 = 35 \cdot 27a^3 \cdot 16b^4 = 15120a^3b^4\)
- Sixth term: \(21(3a)^2(2b)^5 = 21 \cdot 9a^2 \cdot 32b^5 = 6048a^2b^5\)
- Seventh term: \(7(3a)^1(2b)^6 = 7 \cdot 3a \cdot 64b^6 = 1344ab^6\)
- Eighth term: \((3a)^0(2b)^7 = 128b^7\)

4. **Final expanded form:**

\[
(3a + 2b)^7 = 2187a^7 + 10206a^6b + 20412a^5b^2 + 22680a^4b^3 + 15120a^3b^4 + 6048a^2b^5 + 1344ab^6 + 128b^7
\]

This is the complete expansion of \((3a + 2b)^7\).

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Would you like any clarifications or further details on this?

Here are 5 related questions to expand on this topic:
1. How would you expand \((4x + y)^6\) using Pascal's Triangle?
2. What is Pascal's Triangle, and how is it derived?
3. Can you find the expansion of \((2a - b)^5\) using the binomial theorem?
4. How does the binomial theorem apply to negative exponents?
5. How can Pascal's Triangle be used in combinatorics?

**Tip:** Pascal's Triangle can help solve combinatorial problems like finding the number of ways to choose items, as its entries correspond to binomial coefficients.

Explore the step-by-step expansion of the expression (3a + 2b)⁷ using Pascal's Triangle. Learn about binomial expansion, Pascal's Triangle coefficients, and simplifying the final expression. This example is suitable for high school students in Grades 10-12.

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