If 7th term in the binomial expansion of `((3)/(root3(84))+sqrt3ln x)^9, x gt 0` is equal to 729, then x can be

Options

A. `e^(2)`
B. e
C. `(e )/(2)`
D. 2e

Answer

To solve the problem, we need to find the 7th term in the binomial expansion of the expression \(\left(\frac{3}{\sqrt[3]{84}} + \sqrt{3} \ln x\right)^9\) and set it equal to 729. Let's break it down step by step.

### Step 1: Identify the general term in the binomial expansion
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, \(n = 9\), \(a = \frac{3}{\sqrt[3]{84}}\), and \(b = \sqrt{3} \ln x\).

### Step 2: Find the 7th term
To find the 7th term, we need \(r = 6\) (since \(T_{7} = T_{6+1}\)):
\[ T_7 = \binom{9}{6} a^{9-6} b^6 = \binom{9}{6} a^3 b^6 \]

### Step 3: Calculate \(\binom{9}{6}\)
Using the formula for combinations:
\[ \binom{9}{6} = \binom{9}{3} = \frac{9!}{3!6!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]

### Step 4: Substitute \(a\) and \(b\)
Now substituting \(a\) and \(b\):
\[ T_7 = 84 \left(\frac{3}{\sqrt[3]{84}}\right)^3 \left(\sqrt{3} \ln x\right)^6 \]

### Step 5: Simplify \(a^3\)
Calculating \(a^3\):
\[ \left(\frac{3}{\sqrt[3]{84}}\right)^3 = \frac{3^3}{\sqrt[3]{84^3}} = \frac{27}{84} \]

### Step 6: Simplify \(b^6\)
Calculating \(b^6\):
\[ \left(\sqrt{3} \ln x\right)^6 = 3^3 (\ln x)^6 = 27 (\ln x)^6 \]

### Step 7: Combine the terms
Now substituting back:
\[ T_7 = 84 \cdot \frac{27}{84} \cdot 27 (\ln x)^6 \]
The \(84\) cancels out:
\[ T_7 = 27 \cdot 27 (\ln x)^6 = 729 (\ln x)^6 \]

### Step 8: Set the equation equal to 729
We are given that \(T_7 = 729\):
\[ 729 (\ln x)^6 = 729 \]

### Step 9: Solve for \(\ln x\)
Dividing both sides by 729:
\[ (\ln x)^6 = 1 \]
Taking the sixth root:
\[ \ln x = 1 \quad \text{or} \quad \ln x = -1 \]

### Step 10: Solve for \(x\)
Exponentiating both sides:
1. If \(\ln x = 1\), then \(x = e^1 = e\).
2. If \(\ln x = -1\), then \(x = e^{-1} = \frac{1}{e}\).

Since \(x > 0\), both values are valid.

### Final Answer
The possible values of \(x\) are:
\[ x = e \quad \text{or} \quad x = \frac{1}{e} \]

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