The fifth term of `(1 - (2x)/(3))^(3//4)` is

Options

A. `(-5x^4)/(1152)`
B. `(5x^4)/(1152)`
C. ` - (5x^4)/(1052)`
D. `(5x^4)/(1052)`

Answer

To find the fifth term of the expression \((1 - \frac{2x}{3})^{\frac{3}{4}}\), we can use the Binomial Theorem. The Binomial Theorem states that:

\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

In our case, we have \(a = 1\), \(b = -\frac{2x}{3}\), and \(n = \frac{3}{4}\).

### Step 1: Identify the general term
The general term \(T_{k+1}\) in the expansion is given by:

\[ T_{k+1} = \binom{n}{k} a^{n-k} b^k \]

Substituting our values, we get:

\[ T_{k+1} = \binom{\frac{3}{4}}{k} (1)^{\frac{3}{4}-k} \left(-\frac{2x}{3}\right)^k = \binom{\frac{3}{4}}{k} \left(-\frac{2x}{3}\right)^k \]

### Step 2: Find the fifth term
To find the fifth term, we need to set \(k = 4\) (since the first term corresponds to \(k = 0\)):

\[ T_5 = \binom{\frac{3}{4}}{4} \left(-\frac{2x}{3}\right)^4 \]

### Step 3: Calculate the binomial coefficient
The binomial coefficient \(\binom{\frac{3}{4}}{4}\) is calculated as:

\[ \binom{\frac{3}{4}}{4} = \frac{\frac{3}{4} \left(\frac{3}{4} - 1\right) \left(\frac{3}{4} - 2\right) \left(\frac{3}{4} - 3\right)}{4!} \]

Calculating each term:

- \(\frac{3}{4} - 1 = -\frac{1}{4}\)
- \(\frac{3}{4} - 2 = -\frac{5}{4}\)
- \(\frac{3}{4} - 3 = -\frac{9}{4}\)

Thus,

\[ \binom{\frac{3}{4}}{4} = \frac{\frac{3}{4} \cdot -\frac{1}{4} \cdot -\frac{5}{4} \cdot -\frac{9}{4}}{4!} \]

Calculating the numerator:

\[ = \frac{3 \cdot (-1) \cdot 5 \cdot (-9)}{4^4} = \frac{135}{256} \]

And \(4! = 24\), so:

\[ \binom{\frac{3}{4}}{4} = \frac{135}{256 \cdot 24} = \frac{135}{6144} \]

### Step 4: Calculate \(\left(-\frac{2x}{3}\right)^4\)
Now calculate \(\left(-\frac{2x}{3}\right)^4\):

\[ \left(-\frac{2x}{3}\right)^4 = \frac{16x^4}{81} \]

### Step 5: Combine the results
Now substitute back into the term:

\[ T_5 = \frac{135}{6144} \cdot \frac{16x^4}{81} \]

Calculating this gives:

\[ T_5 = \frac{135 \cdot 16}{6144 \cdot 81} x^4 \]

Calculating \(135 \cdot 16 = 2160\) and \(6144 \cdot 81 = 497664\):

\[ T_5 = \frac{2160}{497664} x^4 \]

### Step 6: Simplify the fraction
Now simplify \(\frac{2160}{497664}\):

\[ T_5 = \frac{5}{1152} x^4 \]

Thus, the fifth term of the expansion is:

\[ T_5 = -\frac{5}{1152} x^4 \]

### Final Answer
The fifth term of the expansion \((1 - \frac{2x}{3})^{\frac{3}{4}}\) is:

\[ -\frac{5}{1152} x^4 \]