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Step-by-Step Solution
Step 1: Rewrite the given expression
The original expression is
\left( \frac{x + 1}{x^{2/3} - x^{1/3} + 1} \;-\; \frac{x - 1}{x - x^{1/2}} \right)^{10} .
We aim to simplify the expression inside the parentheses first.
Step 2: Simplify each fraction
Observe that
x^{2/3} - x^{1/3} + 1
can pair with
x^{1/3} + 1
because
(x^{1/3} + 1)(x^{2/3} - x^{1/3} + 1) = x + 1 .
Also,
x - x^{1/2} = x^{1/2}(x^{1/2} - 1) ,
hence
\frac{x - 1}{x - x^{1/2}} = \frac{(x^{1/2} - 1)(x^{1/2} + 1)}{x^{1/2}(x^{1/2} - 1)}
= \frac{x^{1/2} + 1}{x^{1/2}}
= 1 + \frac{1}{\sqrt{x}}.
Step 3: Combine the terms
Using these observations:
\frac{x + 1}{x^{2/3} - x^{1/3} + 1}
= \frac{(x^{1/3} + 1)(x^{2/3} - x^{1/3} + 1)}{x^{2/3} - x^{1/3} + 1}
= x^{1/3} + 1.
Hence,
\left( x^{1/3} + 1 \right) - \left( 1 + \frac{1}{\sqrt{x}} \right)
= x^{1/3} + 1 - 1 - \frac{1}{\sqrt{x}}
= x^{1/3} - \frac{1}{x^{1/2}}.
Step 4: Recognize the binomial form
Therefore, the expression inside the power 10 simplifies to
\left( x^{1/3} - \frac{1}{x^{1/2}} \right).
Our task is to find the term independent of x in
\left( x^{1/3} - \frac{1}{x^{1/2}} \right)^{10}.
Step 5: Write the general term
The general term T_{r+1} in the expansion of
(a - b)^{n}
is given by:
T_{r+1} = \binom{n}{r} \, a^{\,n-r} \, (-b)^{\,r},
which, when translated to our expression, becomes:
T_{r+1} = \binom{10}{r}
\left( x^{1/3} \right)^{10-r}
\left( -\frac{1}{x^{1/2}} \right)^{r}.
Step 6: Determine the exponent of x
In T_{r+1} :
1. The factor (x^{1/3})^{10-r} = x^{\frac{10-r}{3}}.
2. The factor \left( -\frac{1}{x^{1/2}} \right)^{r} = (-1)^{r} \, x^{-\frac{r}{2}}.
So the total power of x in T_{r+1} is
\frac{10 - r}{3} - \frac{r}{2}.
Step 7: Set the total exponent of x to 0
For the term to be independent of x , we require
\frac{10 - r}{3} - \frac{r}{2} = 0.
Solve for r :
\frac{10 - r}{3} = \frac{r}{2}
Cross-multiplying,
2(10 - r) = 3r
20 - 2r = 3r
20 = 5r
r = 4.
Step 8: Compute the independent term
The term independent of x corresponds to r = 4 . This is the 5th term in the expansion (since r+1 = 5 ). The binomial coefficient for that term is:
\binom{10}{4} = \frac{10!}{4!\,6!} = 210.
The factor (-1)^{4} = 1 does not change the magnitude here, so the constant term is
210.
Final Answer
The term independent of x in the given expansion is
210.
