Write out the expansions of the following:
`(3x-y)^(4)`

Answer

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

### Step-by-Step Solution:

1. **Identify the Binomial Theorem Formula**:
The Binomial Theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient.

2. **Assign Values**:
In our case, \(a = 3x\), \(b = -y\), and \(n = 4\).

3. **Write the Expansion**:
Using the Binomial Theorem, we can write:
\[ (3x - y)^4 = \sum_{k=0}^{4} \binom{4}{k} (3x)^{4-k} (-y)^k \]

4. **Calculate Each Term**:
We will compute each term for \(k = 0, 1, 2, 3, 4\):

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

- For \(k = 1\):
\[ \binom{4}{1} (3x)^3 (-y)^1 = 4 \cdot (27x^3) \cdot (-y) = -108x^3y \]

- For \(k = 2\):
\[ \binom{4}{2} (3x)^2 (-y)^2 = 6 \cdot (9x^2) \cdot y^2 = 54x^2y^2 \]

- For \(k = 3\):
\[ \binom{4}{3} (3x)^1 (-y)^3 = 4 \cdot (3x) \cdot (-y^3) = -12xy^3 \]

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

5. **Combine All Terms**:
Now, we combine all the terms we calculated:
\[ (3x - y)^4 = 81x^4 - 108x^3y + 54x^2y^2 - 12xy^3 + y^4 \]

### Final Answer:
The expansion of \((3x - y)^4\) is:
\[ 81x^4 - 108x^3y + 54x^2y^2 - 12xy^3 + y^4 \]