Find the middle terms of `(1/2x+2/3sqrtx)^(11)`

Answer

To find the middle terms of the expression \((\frac{1}{2}x + \frac{2}{3}\sqrt{x})^{11}\), we will follow these steps:

### Step 1: Identify the values of \(a\), \(b\), and \(n\)
In the expression \((a + b)^n\), we have:
- \(a = \frac{1}{2}x\)
- \(b = \frac{2}{3}\sqrt{x}\)
- \(n = 11\)

### Step 2: Determine if \(n\) is odd or even
Since \(n = 11\) is odd, the middle term can be found using the formula for the middle term in the binomial expansion when \(n\) is odd. The middle term is given by the \(\frac{n + 1}{2}\)th term.

### Step 3: Calculate the middle term position
For \(n = 11\):
\[ \text{Middle term position} = \frac{11 + 1}{2} = 6 \]
Thus, the middle term is the 6th term.

### Step 4: Write the general term formula
The general term \(T_{r+1}\) in the binomial expansion is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
For the 6th term, we need \(r = 5\) (since \(r\) starts from 0).

### Step 5: Substitute values into the general term formula
Substituting \(n = 11\), \(r = 5\), \(a = \frac{1}{2}x\), and \(b = \frac{2}{3}\sqrt{x}\):
\[ T_6 = \binom{11}{5} \left(\frac{1}{2}x\right)^{11-5} \left(\frac{2}{3}\sqrt{x}\right)^5 \]

### Step 6: Simplify the expression
Calculating each part:
1. \(\binom{11}{5} = \frac{11!}{5!(11-5)!} = \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2 \times 1} = 462\)
2. \(\left(\frac{1}{2}x\right)^{6} = \frac{1}{2^6}x^6 = \frac{1}{64}x^6\)
3. \(\left(\frac{2}{3}\sqrt{x}\right)^{5} = \left(\frac{2}{3}\right)^{5}(\sqrt{x})^{5} = \frac{32}{243}x^{5/2}\)

### Step 7: Combine the terms
Now, we combine everything:
\[ T_6 = 462 \cdot \frac{1}{64}x^6 \cdot \frac{32}{243}x^{5/2} \]
\[ = 462 \cdot \frac{32}{64 \cdot 243} \cdot x^{6 + \frac{5}{2}} = 462 \cdot \frac{32}{64 \cdot 243} \cdot x^{\frac{12 + 5}{2}} = 462 \cdot \frac{32}{64 \cdot 243} \cdot x^{\frac{17}{2}} \]

### Step 8: Final simplification
Calculating the coefficient:
\[ = 462 \cdot \frac{32}{64 \cdot 243} = 462 \cdot \frac{1}{2 \cdot 243} = \frac{462}{486} = \frac{231}{243} \]
Thus, the middle term is:
\[ \frac{231}{243} x^{\frac{17}{2}} \]

### Conclusion
The middle term of the expression \((\frac{1}{2}x + \frac{2}{3}\sqrt{x})^{11}\) is:
\[ \frac{231}{243} x^{\frac{17}{2}} \]