The term independent of x `(x gt 0 , x ne 1)` in the expansion of `[((x+1))/((x^(2//3) -x^(1//3)+1))-((x-1))/((x-sqrtx))]^10` is

Options

A. 105
B. 210
C. 315
D. 420

Answer

To find the term independent of \( x \) in the expansion of

\[ \left( \frac{x+1}{x^{2/3} - x^{1/3} + 1} - \frac{x-1}{x - \sqrt{x}} \right)^{10}, \]

we will follow these steps:

### Step 1: Simplify the Expression
First, we simplify the expression inside the brackets. We can rewrite the terms separately:

\[ \frac{x+1}{x^{2/3} - x^{1/3} + 1} \quad \text{and} \quad \frac{x-1}{x - \sqrt{x}}. \]

### Step 2: Rationalize Each Term
Let's rationalize each term:

1. **For the first term**:
\[ \frac{x+1}{x^{2/3} - x^{1/3} + 1} = \frac{x^{1/3} + 1}{x^{2/3} - x^{1/3} + 1} = \frac{x^{1/3} + 1}{1 - x^{1/3} + 1} = x^{1/3} + 1 - \frac{1}{x^{1/3}}. \]

2. **For the second term**:
\[ \frac{x-1}{x - \sqrt{x}} = \frac{x-1}{x - x^{1/2}} = \frac{x-1}{x^{1/2}(x^{1/2} - 1)} = \frac{1 - \frac{1}{x}}{1 - \frac{1}{\sqrt{x}}}. \]

### Step 3: Combine the Terms
Now, we can combine the simplified terms:

\[ \left( x^{1/3} + 1 - \frac{1}{x^{1/3}} - (x^{1/2} - 1) \right)^{10}. \]

### Step 4: Further Simplification
This simplifies to:

\[ \left( x^{1/3} - x^{1/2} + 2 - \frac{1}{x^{1/3}} \right)^{10}. \]

### Step 5: Identify the Term Independent of \( x \)
We need to find the term independent of \( x \) in the expansion. The general term in the binomial expansion is given by:

\[ T_r = \binom{10}{r} (x^{1/3})^{10-r} (-x^{1/2})^r. \]

This can be expressed as:

\[ T_r = \binom{10}{r} x^{\frac{10-r}{3}} (-1)^r x^{\frac{r}{2}} = \binom{10}{r} (-1)^r x^{\frac{10 - r}{3} + \frac{r}{2}}. \]

### Step 6: Set the Exponent of \( x \) to Zero
To find the term independent of \( x \), we set the exponent equal to zero:

\[ \frac{10 - r}{3} + \frac{r}{2} = 0. \]

### Step 7: Solve for \( r \)
Multiplying through by 6 to eliminate the fractions:

\[ 2(10 - r) + 3r = 0 \implies 20 - 2r + 3r = 0 \implies r = 20. \]

### Step 8: Find the Coefficient
Now, we substitute \( r = 4 \) into the binomial coefficient:

\[ T_4 = \binom{10}{4} (-1)^4. \]

Calculating \( \binom{10}{4} \):

\[ \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210. \]

### Final Answer
Thus, the term independent of \( x \) in the expansion is:

\[ \boxed{210}. \]