Find the coefficient of
`x^(-6)" in "(3x-(4)/(x))^(10)`

Answer

To find the coefficient of \( x^{-6} \) in the expression \( (3x - \frac{4}{x})^{10} \), we will use the Binomial Theorem. Let's go through the solution step by step.

### Step 1: Write the General Term
The general term \( T_r \) in the expansion of \( (a + b)^n \) is given by:
\[ T_r = \binom{n}{r} a^{n-r} b^r \]
In our case, \( a = 3x \), \( b = -\frac{4}{x} \), and \( n = 10 \). Therefore, the general term becomes:
\[ T_r = \binom{10}{r} (3x)^{10-r} \left(-\frac{4}{x}\right)^r \]

### Step 2: Simplify the General Term
Now, we simplify \( T_r \):
\[ T_r = \binom{10}{r} (3^{10-r} x^{10-r}) \left(-4^r \cdot x^{-r}\right) \]
\[ = \binom{10}{r} 3^{10-r} (-4)^r x^{10-r-r} \]
\[ = \binom{10}{r} 3^{10-r} (-4)^r x^{10-2r} \]

### Step 3: Set the Power of \( x \) to -6
We need the power of \( x \) to be \( -6 \):
\[ 10 - 2r = -6 \]
Solving for \( r \):
\[ 10 + 6 = 2r \implies 16 = 2r \implies r = 8 \]

### Step 4: Substitute \( r \) into the General Term
Now, we substitute \( r = 8 \) into the general term:
\[ T_8 = \binom{10}{8} 3^{10-8} (-4)^8 x^{10-2 \cdot 8} \]
\[ = \binom{10}{8} 3^2 (-4)^8 x^{-6} \]
\[ = \binom{10}{2} 3^2 (-4)^8 x^{-6} \]

### Step 5: Calculate the Coefficient
Now, we calculate the coefficient:
\[ \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45 \]
\[ 3^2 = 9 \]
\[ (-4)^8 = 4^8 = (2^2)^8 = 2^{16} \]
Thus, the coefficient of \( x^{-6} \) is:
\[ \text{Coefficient} = 45 \times 9 \times 2^{16} \]
\[ = 405 \times 2^{16} \]

### Final Answer
The coefficient of \( x^{-6} \) in \( (3x - \frac{4}{x})^{10} \) is:
\[ 405 \times 65536 = 26,553,600 \]