Find the specified term of the expression in each of the following binomials:
(iii) Middle term of `(2 x - (1)/(y) )^(8)`.

Answer

To find the middle term of the expression \((2x - \frac{1}{y})^8\), we can follow these steps:

### Step 1: Identify \(n\)
In the expression \((2x - \frac{1}{y})^8\), the exponent \(n\) is 8.

**Hint:** The exponent in the binomial expression gives you the value of \(n\).

### Step 2: Calculate the total number of terms
The total number of terms in the expansion of a binomial expression is given by \(n + 1\). Therefore, we have:
\[ \text{Total number of terms} = n + 1 = 8 + 1 = 9 \]

**Hint:** Remember that the total number of terms is always one more than the exponent.

### Step 3: Determine the middle term
Since there are 9 terms, the middle term will be the 5th term (as the middle term is given by \(\frac{n + 1}{2}\)).

**Hint:** For an odd number of terms, the middle term is the term at position \(\frac{n + 1}{2}\).

### Step 4: Use the formula for the general term
The general term \(T_{r+1}\) in the expansion of \((a + b)^n\) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
In our case, \(a = 2x\), \(b = -\frac{1}{y}\), and \(n = 8\). We need to find the 5th term, which corresponds to \(r = 4\) (since \(T_{5} = T_{r+1}\)).

**Hint:** The term number corresponds to \(r + 1\), so adjust \(r\) accordingly.

### Step 5: Substitute values into the formula
Now, we can substitute \(n = 8\), \(r = 4\), \(a = 2x\), and \(b = -\frac{1}{y}\) into the formula:
\[ T_5 = \binom{8}{4} (2x)^{8-4} \left(-\frac{1}{y}\right)^4 \]

### Step 6: Calculate each component
1. Calculate \(\binom{8}{4}\):
\[ \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]

2. Calculate \((2x)^4\):
\[ (2x)^4 = 2^4 \cdot x^4 = 16x^4 \]

3. Calculate \(\left(-\frac{1}{y}\right)^4\):
\[ \left(-\frac{1}{y}\right)^4 = \frac{1}{y^4} \]

### Step 7: Combine all components
Now, substitute these values back into the term:
\[ T_5 = 70 \cdot 16x^4 \cdot \frac{1}{y^4} = 1120 \cdot \frac{x^4}{y^4} \]

### Final Answer
Thus, the middle term of the expression \((2x - \frac{1}{y})^8\) is:
\[ \frac{1120x^4}{y^4} \]

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