The value of the numerically greatest term in the expansion of `(4-3x)^(7)` when `x=(2)/(3)` is equal to

Options

A. 71680
B. 35840
C. 10752
D. 86016

Answer

To find the value of the numerically greatest term in the expansion of \((4 - 3x)^7\) when \(x = \frac{2}{3}\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term \(T_{r+1}\) in the binomial expansion of \((a + b)^n\) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
Here, \(n = 7\), \(a = 4\), and \(b = -3x\). Thus, the general term becomes:
\[ T_{r+1} = \binom{7}{r} 4^{7-r} (-3x)^r \]

2. **Set Up the Ratio for Greatest Term**:
To find the numerically greatest term, we need to consider the ratio of consecutive terms:
\[ \left| \frac{T_{r+1}}{T_r} \right| \geq 1 \]
This leads to:
\[ \left| \frac{\binom{7}{r} 4^{7-r} (-3x)^r}{\binom{7}{r-1} 4^{7-(r-1)} (-3x)^{r-1}} \right| \geq 1 \]

3. **Simplify the Ratio**:
Simplifying the ratio gives:
\[ \left| \frac{7 - r + 1}{r} \cdot \frac{-3x}{4} \right| \geq 1 \]
This can be rewritten as:
\[ \left| \frac{-3x}{4} \cdot \frac{8 - r}{r} \right| \geq 1 \]

4. **Substitute \(x = \frac{2}{3}\)**:
Substitute \(x\) into the inequality:
\[ \left| \frac{-3 \cdot \frac{2}{3}}{4} \cdot \frac{8 - r}{r} \right| = \left| \frac{-2}{4} \cdot \frac{8 - r}{r} \right| = \left| \frac{1}{2} \cdot \frac{8 - r}{r} \right| \geq 1 \]
This implies:
\[ \frac{8 - r}{r} \geq 2 \]

5. **Solve the Inequality**:
Rearranging gives:
\[ 8 - r \geq 2r \implies 8 \geq 3r \implies r \leq \frac{8}{3} \approx 2.67 \]
Since \(r\) must be an integer, the maximum integer value for \(r\) is \(2\).

6. **Find the Greatest Term**:
Now, we calculate the third term \(T_3\) (where \(r = 2\)):
\[ T_3 = \binom{7}{2} 4^{7-2} (-3x)^2 \]
Substitute \(x = \frac{2}{3}\):
\[ T_3 = \binom{7}{2} 4^5 (-3 \cdot \frac{2}{3})^2 \]
Calculate:
\[ T_3 = 21 \cdot 1024 \cdot 4 = 21 \cdot 4096 \]

7. **Final Calculation**:
\[ T_3 = 21 \cdot 4096 = 86016 \]

### Conclusion:
The value of the numerically greatest term in the expansion of \((4 - 3x)^7\) when \(x = \frac{2}{3}\) is **86016**.