If the fourth term in the binomial expansion of `((2)/(x) + x^(log_(8) x))^(6) (x gt 0)` is `20 xx 8^(7)`, then the value of `x` is

Options

A. `8^(-2)`
B. `8^(3)`
C. 8
D. `8^(2)`

Explanation

To solve the problem, we need to find the value of \( x \) given that the fourth term in the binomial expansion of

\[ \left(\frac{2}{x} + x^{\log_8 x}\right)^6 \]

is equal to \( 20 \times 8^7 \).

### Step-by-step Solution:

1. **Identify the General Term**:
The general term \( T_{r+1} \) in the binomial expansion is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
where \( n = 6 \), \( a = \frac{2}{x} \), and \( b = x^{\log_8 x} \).

2. **Write the Fourth Term**:
The fourth term corresponds to \( r = 3 \):
\[ T_4 = \binom{6}{3} \left(\frac{2}{x}\right)^{6-3} \left(x^{\log_8 x}\right)^3 \]
Simplifying this gives:
\[ T_4 = \binom{6}{3} \left(\frac{2}{x}\right)^3 \left(x^{\log_8 x}\right)^3 \]

3. **Calculate \( \binom{6}{3} \)**:
\[ \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

4. **Substituting Values**:
Substitute \( \binom{6}{3} \) into the term:
\[ T_4 = 20 \cdot \left(\frac{2}{x}\right)^3 \cdot \left(x^{\log_8 x}\right)^3 = 20 \cdot \frac{8}{x^3} \cdot x^{3 \log_8 x} \]
This simplifies to:
\[ T_4 = 20 \cdot \frac{8}{x^3} \cdot x^{3 \log_8 x} = 20 \cdot 8 \cdot x^{3 \log_8 x - 3} \]

5. **Set Equal to Given Value**:
We know that:
\[ T_4 = 20 \times 8^7 \]
Therefore, we set:
\[ 20 \cdot 8 \cdot x^{3 \log_8 x - 3} = 20 \cdot 8^7 \]

6. **Cancel Common Terms**:
Dividing both sides by \( 20 \):
\[ 8 \cdot x^{3 \log_8 x - 3} = 8^7 \]
Dividing both sides by \( 8 \):
\[ x^{3 \log_8 x - 3} = 8^6 \]

7. **Express \( 8^6 \) in Terms of Base 8**:
Since \( 8 = 2^3 \), we can write:
\[ 8^6 = (2^3)^6 = 2^{18} = 8^6 \]

8. **Taking Logarithm**:
Taking logarithm base 8 on both sides:
\[ 3 \log_8 x - 3 = 6 \]
Rearranging gives:
\[ 3 \log_8 x = 9 \quad \Rightarrow \quad \log_8 x = 3 \]

9. **Solving for \( x \)**:
Converting back from logarithmic form:
\[ x = 8^3 = 512 \]

### Final Answer:
Thus, the value of \( x \) is \( \boxed{512} \).