Find the term independent of `x` in the expansion of `((3)/(2) x^(2) - (1)/(3x) )^(9)`.

Answer

To find the term independent of \( x \) in the expansion of \( \left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^9 \), we will follow these steps:

### Step 1: Identify the general term in the binomial expansion
The general term \( T_{r+1} \) in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
In our case, \( n = 9 \), \( a = \frac{3}{2} x^2 \), and \( b = -\frac{1}{3x} \).

### Step 2: Write the general term for our specific case
Substituting the values into the formula, we get:
\[ T_{r+1} = \binom{9}{r} \left( \frac{3}{2} x^2 \right)^{9-r} \left( -\frac{1}{3x} \right)^r \]

### Step 3: Simplify the general term
Expanding this, we have:
\[ T_{r+1} = \binom{9}{r} \left( \frac{3}{2} \right)^{9-r} (x^2)^{9-r} \left( -\frac{1}{3} \right)^r (x^{-1})^r \]
This simplifies to:
\[ T_{r+1} = \binom{9}{r} \left( \frac{3}{2} \right)^{9-r} \left( -\frac{1}{3} \right)^r x^{2(9-r) - r} \]
\[ = \binom{9}{r} \left( \frac{3}{2} \right)^{9-r} \left( -\frac{1}{3} \right)^r x^{18 - 3r} \]

### Step 4: Find the term independent of \( x \)
For the term to be independent of \( x \), the exponent of \( x \) must be zero:
\[ 18 - 3r = 0 \]
Solving for \( r \):
\[ 3r = 18 \implies r = 6 \]

### Step 5: Substitute \( r \) back to find the specific term
Now, we find the term \( T_{r+1} \) when \( r = 6 \):
\[ T_{7} = \binom{9}{6} \left( \frac{3}{2} \right)^{3} \left( -\frac{1}{3} \right)^{6} \]

### Step 6: Calculate the binomial coefficient and powers
Calculating \( \binom{9}{6} \):
\[ \binom{9}{6} = \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]
Calculating \( \left( \frac{3}{2} \right)^{3} \):
\[ \left( \frac{3}{2} \right)^{3} = \frac{27}{8} \]
Calculating \( \left( -\frac{1}{3} \right)^{6} \):
\[ \left( -\frac{1}{3} \right)^{6} = \frac{1}{729} \]

### Step 7: Combine all parts to find the term
Now substituting these values back:
\[ T_{7} = 84 \cdot \frac{27}{8} \cdot \frac{1}{729} \]
Calculating:
\[ T_{7} = \frac{84 \cdot 27}{8 \cdot 729} \]
\[ = \frac{2268}{5832} = \frac{7}{18} \]

### Final Answer
The term independent of \( x \) in the expansion is:
\[ \frac{7}{18} \]