Expand `(3x-2y)^(4)`.

Answer

To expand \((3x - 2y)^4\) using the Binomial Theorem, we follow these steps:

### Step 1: Identify \(a\), \(b\), and \(n\)
In the expression \((3x - 2y)^4\):
- Let \(a = 3x\)
- Let \(b = -2y\)
- Let \(n = 4\)

### Step 2: Apply the Binomial Theorem
The Binomial Theorem states that:
\[ (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \]
For our case, we will use:
\[ (3x - 2y)^4 = \sum_{r=0}^{4} \binom{4}{r} (3x)^{4-r} (-2y)^r \]

### Step 3: Calculate each term in the expansion
We will calculate each term for \(r = 0\) to \(r = 4\):

1. **For \(r = 0\)**:
\[ \binom{4}{0} (3x)^{4} (-2y)^{0} = 1 \cdot (3x)^4 \cdot 1 = 81x^4 \]

2. **For \(r = 1\)**:
\[ \binom{4}{1} (3x)^{3} (-2y)^{1} = 4 \cdot (3x)^3 \cdot (-2y) = 4 \cdot 27x^3 \cdot (-2y) = -216x^3y \]

3. **For \(r = 2\)**:
\[ \binom{4}{2} (3x)^{2} (-2y)^{2} = 6 \cdot (3x)^2 \cdot (4y^2) = 6 \cdot 9x^2 \cdot 4y^2 = 216x^2y^2 \]

4. **For \(r = 3\)**:
\[ \binom{4}{3} (3x)^{1} (-2y)^{3} = 4 \cdot (3x) \cdot (-8y^3) = 4 \cdot 3x \cdot (-8y^3) = -96xy^3 \]

5. **For \(r = 4\)**:
\[ \binom{4}{4} (3x)^{0} (-2y)^{4} = 1 \cdot 1 \cdot 16y^4 = 16y^4 \]

### Step 4: Combine all the terms
Now, we combine all the terms we calculated:
\[ (3x - 2y)^4 = 81x^4 - 216x^3y + 216x^2y^2 - 96xy^3 + 16y^4 \]

### Final Answer
Thus, the expansion of \((3x - 2y)^4\) is:
\[ 81x^4 - 216x^3y + 216x^2y^2 - 96xy^3 + 16y^4 \]

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