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

Answer

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

### Step 1: Understand the Binomial Theorem
The binomial theorem states that:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
In our case, \(a = 3x\), \(b = -2y\), and \(n = 4\).

### Step 2: Write the Expansion
Using the binomial theorem, we can expand \((3x - 2y)^4\) as follows:
\[ (3x - 2y)^4 = \sum_{k=0}^{4} \binom{4}{k} (3x)^{4-k} (-2y)^k \]

### Step 3: Calculate Each Term
Now we will calculate each term in the expansion for \(k = 0\) to \(k = 4\):

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

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

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

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

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

### Step 4: Combine All 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:
\[ 81x^4 - 216x^3y + 216x^2y^2 - 96xy^3 + 16y^4 \]

---