Write out the expansions of the following:
`(3+2x^(2) )^(4)`

Answer

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

### Step 1: Identify the components
In the expression \( (3 + 2x^2)^4 \):
- \( a = 3 \)
- \( b = 2x^2 \)
- \( n = 4 \)

### Step 2: Apply the Binomial Theorem
The Binomial Theorem states that:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
For our expression, we can write:
\[ (3 + 2x^2)^4 = \sum_{k=0}^{4} \binom{4}{k} (3)^{4-k} (2x^2)^k \]

### Step 3: Expand the summation
Now we will calculate each term in the summation for \( k = 0, 1, 2, 3, 4 \):

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

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

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

- **For \( k = 3 \)**:
\[ \binom{4}{3} (3)^{4-3} (2x^2)^3 = 4 \cdot 3 \cdot 8x^6 = 96x^6 \]

- **For \( k = 4 \)**:
\[ \binom{4}{4} (3)^{4-4} (2x^2)^4 = 1 \cdot 1 \cdot 16x^8 = 16x^8 \]

### Step 4: Combine all terms
Now, we combine all the terms we calculated:
\[ (3 + 2x^2)^4 = 81 + 216x^2 + 216x^4 + 96x^6 + 16x^8 \]

### Final Result
Thus, the expansion of \( (3 + 2x^2)^4 \) is:
\[ \boxed{81 + 216x^2 + 216x^4 + 96x^6 + 16x^8} \]