If the 6th term in the expansion of `((1)/(x^(8//3))+x^(2)log_(10)x)^(8)` is 5600, then x equals:

Options

A. 2
B. `sqrt(5)`
C. `sqrt(10)`
D. 10

Explanation

To solve the problem, we need to find the value of \( x \) given that the 6th term in the expansion of \( \left( \frac{1}{x^{8/3}} + x^2 \log_{10} x \right)^8 \) is equal to 5600.

### 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, \( n = 8 \), \( a = \frac{1}{x^{8/3}} \), and \( b = x^2 \log_{10} x \).

2. **Find the 6th Term**:
The 6th term corresponds to \( r = 5 \) (since \( T_{r+1} \) corresponds to \( r \)):
\[ T_6 = \binom{8}{5} \left( \frac{1}{x^{8/3}} \right)^{8-5} \left( x^2 \log_{10} x \right)^5 \]

3. **Calculate the Binomial Coefficient**:
\[ \binom{8}{5} = \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]

4. **Substitute into the Term**:
Substitute the values into the expression for \( T_6 \):
\[ T_6 = 56 \left( \frac{1}{x^{8/3}} \right)^{3} \left( x^2 \log_{10} x \right)^5 \]
\[ = 56 \cdot \frac{1}{x^{8}} \cdot (x^2)^5 \cdot (\log_{10} x)^5 \]
\[ = 56 \cdot \frac{x^{10}}{x^{8}} \cdot (\log_{10} x)^5 = 56 \cdot x^{2} \cdot (\log_{10} x)^5 \]

5. **Set the Equation Equal to 5600**:
We know that this term equals 5600:
\[ 56 \cdot x^{2} \cdot (\log_{10} x)^5 = 5600 \]

6. **Simplify the Equation**:
Divide both sides by 56:
\[ x^{2} \cdot (\log_{10} x)^5 = \frac{5600}{56} = 100 \]

7. **Rearranging the Equation**:
We can express this as:
\[ x^{2} = \frac{100}{(\log_{10} x)^5} \]

8. **Assume \( \log_{10} x = y \)**:
Then \( x = 10^y \) and substituting gives:
\[ (10^y)^{2} = \frac{100}{y^5} \]
\[ 10^{2y} = \frac{100}{y^5} \]
\[ 10^{2y} = 10^2 \cdot y^{-5} \]

9. **Equate the Exponents**:
Since the bases are the same, we can equate the exponents:
\[ 2y = 2 - 5 \log_{10} y \]

10. **Solve for \( y \)**:
Rearranging gives:
\[ 5 \log_{10} y + 2y - 2 = 0 \]
This is a transcendental equation and may require numerical methods or graphing to solve.

11. **Final Calculation**:
By testing values, we find that \( y = 1 \) (i.e., \( \log_{10} x = 1 \)) gives:
\[ x = 10 \]

### Conclusion:
Thus, the value of \( x \) is:
\[ \boxed{10} \]