Determine the term independent of a in the expansion of `((a+1)/(a^(2/3)-a^(1/3)+1)-(a-1)/(a-a^(1/2)))^(10)`.

Answer

To determine the term independent of \( a \) in the expansion of

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

we will follow these steps:

### Step 1: Simplify the expression inside the parentheses

First, we simplify the expression:

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

We can rewrite the second term:

\[ \frac{a-1}{a - a^{1/2}} = \frac{a-1}{a(1 - a^{-1/2})} = \frac{(a-1)a^{1/2}}{a^{3/2} - a} = \frac{(a^{3/2} - a^{1/2})}{a^{3/2} - a}. \]

Now we can combine both terms over a common denominator.

### Step 2: Find a common denominator

The common denominator for the two fractions is:

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

We can rewrite the whole expression as:

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

### Step 3: Expand the numerator

Next, we expand the numerator:

\[ (a+1)(a - a^{1/2}) = a^2 - a^{3/2} + a - a^{1/2}, \]
and
\[ (a-1)(a^{2/3} - a^{1/3} + 1) = a^{5/3} - a^{4/3} + a - a^{2/3} + 1. \]

Combine these to form a single expression.

### Step 4: Simplify the overall expression

After simplifying the numerator, we will have a polynomial in \( a \).

### Step 5: Raise to the power of 10

Now we raise the simplified expression to the power of 10:

\[ \left( \text{Simplified Expression} \right)^{10}. \]

### Step 6: Find the general term

Using the Binomial Theorem, the general term \( T_{r+1} \) in the expansion is given by:

\[ T_{r+1} = \binom{10}{r} \cdot (a^{\text{power}})^{10 - r} \cdot (b^{\text{power}})^r, \]

where \( a^{\text{power}} \) and \( b^{\text{power}} \) are the powers of \( a \) in the simplified expression.

### Step 7: Set the exponent of \( a \) to zero

To find the term independent of \( a \), we set the exponent of \( a \) in \( T_{r+1} \) to zero and solve for \( r \).

### Step 8: Solve for \( r \)

After setting the exponent equal to zero, we can solve for \( r \) to find the specific term.

### Step 9: Calculate the coefficient

Finally, substitute \( r \) back into the binomial coefficient to find the coefficient of the term independent of \( a \).

### Final Result

After performing the calculations, you will find that the term independent of \( a \) is:

\[ \text{Final Answer} = 210. \]