`((x+1)/(x^(2/3)-x^(1/3)+1)-(x-1)/(x+x^(1/2)))^10` find term independent of x

Answer

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

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

we will follow these steps:

### Step 1: Simplify the Expression
We start by simplifying the expression inside the parentheses.

1. **First Term**:
\[ \frac{x+1}{x^{2/3} - x^{1/3} + 1} \]
can be rewritten as:
\[ \frac{x^{1/3} + 1}{x^{2/3} - x^{1/3} + 1} = \frac{x^{1/3} + 1}{(x^{1/3})^2 - x^{1/3} + 1} \]

2. **Second Term**:
\[ \frac{x-1}{x + x^{1/2}} = \frac{x^{1/2}(x^{1/2} - 1)}{x^{1/2}(x^{1/2} + 1)} = \frac{x^{1/2} - 1}{x^{1/2} + 1} \]

Now, we can combine these two fractions.

### Step 2: Combine the Two Fractions
To combine the two fractions, we find a common denominator:

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

### Step 3: Expand and Simplify
We will expand the numerator and simplify it. This will involve multiplying out the terms and combining like terms.

### Step 4: Identify the Binomial Expansion
After simplification, we will have a polynomial in \( x \). We will raise this polynomial to the power of 10.

### Step 5: Find the General Term
The general term of a binomial expansion \( (a + b)^n \) is given by:

\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]

In our case, we will identify \( a \) and \( b \) from our simplified expression and find the general term.

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

\[ \text{Exponent of } x = 0 \]

### Step 7: Solve for \( r \)
From the equation obtained in Step 6, we will solve for \( r \) to find the specific term that is independent of \( x \).

### Step 8: Calculate the Coefficient
Once we have \( r \), we will substitute it back into the general term to find the coefficient of the term that is independent of \( x \).

### Step 9: Conclusion
Finally, we will conclude with the value of the coefficient which represents the term independent of \( x \).