`((x^(5y-3)xx x^(3-2y))/(x^(4y-6)xx x^(2y-9)))^(-4/3)=`______

Options

A. `x^(3y+15)`
B. `x^(13-3y)`
C. `x^(4y-20)`
D. `x^(4y+18)`

Explanation

To solve the expression \(\left(\frac{x^{(5y-3) \cdot x^{(3-2y)}}{x^{(4y-6) \cdot x^{(2y-9)}}}\right)^{-\frac{4}{3}}\), we will simplify it step by step using the rules of indices.

### Step-by-Step Solution:

1. **Combine the powers in the numerator and denominator:**
\[ \text{Numerator: } x^{(5y-3) + (3-2y)} = x^{(5y - 3 + 3 - 2y)} = x^{(5y - 2y)} = x^{3y} \]
\[ \text{Denominator: } x^{(4y-6) + (2y-9)} = x^{(4y - 6 + 2y - 9)} = x^{(6y - 15)} \]

2. **Rewrite the expression:**
\[ \left(\frac{x^{3y}}{x^{6y-15}}\right)^{-\frac{4}{3}} \]

3. **Apply the quotient rule of indices:**
\[ x^{3y - (6y - 15)} = x^{3y - 6y + 15} = x^{-3y + 15} \]

4. **Now apply the power of a power rule:**
\[ \left(x^{-3y + 15}\right)^{-\frac{4}{3}} = x^{(-3y + 15) \cdot -\frac{4}{3}} = x^{\frac{4(3y - 15)}{3}} = x^{\frac{12y - 60}{3}} = x^{4y - 20} \]

5. **Final answer:**
\[ x^{4y - 20} \]