Write out the expansion of the following: `(2x + (y)/(2) )^(5)`

Answer

To expand the expression \((2x + \frac{y}{2})^5\) using the Binomial Theorem, we follow these steps:

### Step 1: Identify the terms
In the expression \((a + b)^n\), we have:
- \(a = 2x\)
- \(b = \frac{y}{2}\)
- \(n = 5\)

### 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, it becomes:
\[ (2x + \frac{y}{2})^5 = \sum_{r=0}^{5} \binom{5}{r} (2x)^{5-r} \left(\frac{y}{2}\right)^r \]

### Step 3: Write out the terms
Now we will calculate each term for \(r = 0\) to \(r = 5\):

1. **For \(r = 0\)**:
\[ \binom{5}{0} (2x)^5 \left(\frac{y}{2}\right)^0 = 1 \cdot (2x)^5 \cdot 1 = 32x^5 \]

2. **For \(r = 1\)**:
\[ \binom{5}{1} (2x)^4 \left(\frac{y}{2}\right)^1 = 5 \cdot (2x)^4 \cdot \frac{y}{2} = 5 \cdot 16x^4 \cdot \frac{y}{2} = 40x^4y \]

3. **For \(r = 2\)**:
\[ \binom{5}{2} (2x)^3 \left(\frac{y}{2}\right)^2 = 10 \cdot (2x)^3 \cdot \left(\frac{y^2}{4}\right) = 10 \cdot 8x^3 \cdot \frac{y^2}{4} = 20x^3y^2 \]

4. **For \(r = 3\)**:
\[ \binom{5}{3} (2x)^2 \left(\frac{y}{2}\right)^3 = 10 \cdot (2x)^2 \cdot \left(\frac{y^3}{8}\right) = 10 \cdot 4x^2 \cdot \frac{y^3}{8} = 5x^2y^3 \]

5. **For \(r = 4\)**:
\[ \binom{5}{4} (2x)^1 \left(\frac{y}{2}\right)^4 = 5 \cdot (2x) \cdot \left(\frac{y^4}{16}\right) = 5 \cdot 2x \cdot \frac{y^4}{16} = \frac{5xy^4}{8} \]

6. **For \(r = 5\)**:
\[ \binom{5}{5} (2x)^0 \left(\frac{y}{2}\right)^5 = 1 \cdot 1 \cdot \frac{y^5}{32} = \frac{y^5}{32} \]

### Step 4: Combine all terms
Now, we combine all the terms we calculated:
\[ (2x + \frac{y}{2})^5 = 32x^5 + 40x^4y + 20x^3y^2 + 5x^2y^3 + \frac{5xy^4}{8} + \frac{y^5}{32} \]

### Final Answer:
The expansion of \((2x + \frac{y}{2})^5\) is:
\[ 32x^5 + 40x^4y + 20x^3y^2 + 5x^2y^3 + \frac{5xy^4}{8} + \frac{y^5}{32} \]