Find the term independent of x in `((x^(1//2))/(3) - (4)/(x^2))^10`

Answer

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

\[ \left( \frac{\sqrt{x}}{3} - \frac{4}{x^2} \right)^{10}, \]

we can use the Binomial Theorem. The general term in the expansion of \( (a + b)^n \) is given by:

\[ T_r = \binom{n}{r} a^{n-r} b^r, \]

where \( a = \frac{\sqrt{x}}{3} \) and \( b = -\frac{4}{x^2} \), and \( n = 10 \).

### Step 1: Identify the general term
The general term \( T_r \) can be expressed as:

\[ T_r = \binom{10}{r} \left( \frac{\sqrt{x}}{3} \right)^{10-r} \left( -\frac{4}{x^2} \right)^r. \]

### Step 2: Simplify the general term
Now, simplifying \( T_r \):

\[ T_r = \binom{10}{r} \left( \frac{x^{1/2}}{3} \right)^{10-r} \left( -\frac{4}{x^2} \right)^r = \binom{10}{r} \frac{(-4)^r}{3^{10-r}} x^{\frac{10-r}{2} - 2r}. \]

### Step 3: Find the exponent of \( x \)
The exponent of \( x \) in \( T_r \) is:

\[ \frac{10 - r}{2} - 2r = \frac{10 - r - 4r}{2} = \frac{10 - 5r}{2}. \]

### Step 4: Set the exponent to zero
To find the term independent of \( x \), we set the exponent equal to zero:

\[ \frac{10 - 5r}{2} = 0. \]

Multiplying through by 2 gives:

\[ 10 - 5r = 0 \implies 5r = 10 \implies r = 2. \]

### Step 5: Substitute \( r \) back into the general term
Now we substitute \( r = 2 \) back into the general term \( T_r \):

\[ T_2 = \binom{10}{2} \left( \frac{\sqrt{x}}{3} \right)^{10-2} \left( -\frac{4}{x^2} \right)^2. \]

Calculating \( T_2 \):

\[ T_2 = \binom{10}{2} \left( \frac{\sqrt{x}}{3} \right)^{8} \left( -\frac{4}{x^2} \right)^{2}. \]

### Step 6: Calculate the coefficient
Calculating the binomial coefficient:

\[ \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45. \]

Now substituting:

\[ T_2 = 45 \cdot \left( \frac{x^{4}}{3^8} \right) \cdot \left( \frac{16}{x^4} \right) = 45 \cdot \frac{16}{3^8}. \]

### Step 7: Final calculation
Now we simplify:

\[ T_2 = \frac{720}{6561}. \]

Thus, the term independent of \( x \) is:

\[ \frac{720}{6561}. \]

### Summary
The term independent of \( x \) in the expansion of \( \left( \frac{\sqrt{x}}{3} - \frac{4}{x^2} \right)^{10} \) is

\[ \frac{720}{6561}. \]