If `[(x+1)/(x^(2/3)+1-x^(1/3))-(x-1)/(x+sqrtx)]^10` then find the independent of `x`

Answer

To solve the problem, we need to simplify the expression

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

and find the independent term of \(x\).

### Step 1: Simplify the first fraction

The first fraction is

\[ \frac{x+1}{x^{2/3} + 1 - x^{1/3}}. \]

We can rewrite the denominator using the identity for the sum of cubes:

\[ x^{2/3} + 1 - x^{1/3} = (x^{1/3})^2 + 1 - (x^{1/3}) = (x^{1/3} + 1)((x^{1/3})^2 - (x^{1/3}) + 1). \]

Thus, we have:

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

### Step 2: Simplify the second fraction

The second fraction is

\[ \frac{x-1}{x+\sqrt{x}}. \]

We can factor the denominator:

\[ x + \sqrt{x} = \sqrt{x}(\sqrt{x} + 1). \]

Thus, we rewrite the second fraction as:

\[ \frac{x-1}{\sqrt{x}(\sqrt{x}+1)}. \]

### Step 3: Combine the two fractions

Now we have:

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

To combine these fractions, we need a common denominator, which is:

\[ (x^{1/3}+1)((x^{1/3})^2 - (x^{1/3}) + 1)\sqrt{x}(\sqrt{x}+1). \]

### Step 4: Find the independent term of \(x\)

After combining the fractions, we will have a polynomial expression in terms of \(x\). We need to find the term that does not depend on \(x\) when raised to the power of 10.

Using the binomial theorem, we can express the combined fraction as:

\[ \left[a x^{1/3} + b x^{-1/2}\right]^{10}, \]

where \(a\) and \(b\) are coefficients from the combined fraction.

### Step 5: Use the binomial theorem

Using the binomial theorem, we can express this as:

\[ \sum_{r=0}^{10} \binom{10}{r} (a x^{1/3})^{10-r} (b x^{-1/2})^r. \]

The exponent of \(x\) in each term is given by:

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

Setting this equal to zero to find the independent term:

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

### Step 6: Solve for \(r\)

Multiplying through by 6 to eliminate the denominators:

\[ 2(10 - r) - 3r = 0 \implies 20 - 2r - 3r = 0 \implies 20 = 5r \implies r = 4. \]

### Step 7: Calculate the coefficient

Now we substitute \(r = 4\) back into the binomial coefficient:

\[ \binom{10}{4} a^{6} b^{4}. \]

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

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

### Final Answer

Thus, the independent term of \(x\) in the expression raised to the power of 10 is:

\[ \boxed{210}. \]