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

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

### 1. Coefficients from Pascal’s Triangle:
The coefficients of the expansion come from the 7th row of Pascal’s Triangle, which is:
\[
1, 7, 21, 35, 35, 21, 7, 1
\]

### 2. General form of binomial expansion:
The binomial expansion of \((x + y)^n\) is:
\[
(x + y)^n = \sum_{k=0}^{n} C(n,k) \cdot x^{n-k} \cdot y^k
\]
Where \( C(n,k) \) are the binomial coefficients, which are the numbers from Pascal’s Triangle.

### 3. Apply to \((3a + 2b)^7\):
Here, \(x = 3a\), \(y = 2b\), and \(n = 7\).

Now we’ll apply the coefficients from Pascal’s Triangle and the terms \((3a)\) and \((2b)\) raised to the appropriate powers:

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

### 4. Simplify the terms:
Now simplify each term:

- \((3a)^7 = 2187a^7\)
- \((3a)^6 \cdot (2b) = 1458a^6b\)
- \((3a)^5 \cdot (2b)^2 = 486a^5b^2\)
- \((3a)^4 \cdot (2b)^3 = 864a^4b^3\)
- \((3a)^3 \cdot (2b)^4 = 864a^3b^4\)
- \((3a)^2 \cdot (2b)^5 = 486a^2b^5\)
- \((3a) \cdot (2b)^6 = 1458ab^6\)
- \((2b)^7 = 128b^7\)

### 5. Final expansion:
Now combine all the terms with their respective coefficients:

\[
(3a + 2b)^7 = 2187a^7 + 10206a^6b + 10206a^5b^2 + 30240a^4b^3 + 30240a^3b^4 + 10206a^2b^5 + 10206ab^6 + 128b^7
\]

This is the expanded form of \((3a + 2b)^7\).

Would you like more details or have any questions?

### Related Questions:
1. How is Pascal's Triangle constructed?
2. How do you calculate binomial coefficients without Pascal’s Triangle?
3. Can you apply the binomial theorem to negative exponents?
4. What happens if \( n \) is a fraction in binomial expansions?
5. How does binomial expansion work for larger powers like \( (3a + 2b)^{10} \)?

**Tip**: When using binomial expansions, always ensure the terms are in the form \( (x + y)^n \) for proper application of Pascal's Triangle.

Learn how to expand the binomial expression (3a + 2b)⁷ using Pascal's Triangle. This detailed step-by-step guide covers binomial expansion, Pascal's Triangle coefficients, and algebraic simplifications. Suitable for Grades 10-12 students studying advanced algebra.

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