The expression `(p^(-2x)q^(3y))^(6)div(p^(3)q^(-1))^(-4x)` after simplification becomes

Options

A. independent of p, but not of q
B. independent of q, but not of p
C. independent of both p and q
D. dependent on both p and q but independent of x and y.

Answer

To simplify the expression \((p^{-2x}q^{3y})^{6} \div (p^{3}q^{-1})^{-4x}\), we will follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the expression in a clearer form:
\[ \frac{(p^{-2x}q^{3y})^{6}}{(p^{3}q^{-1})^{-4x}} \]

### Step 2: Apply the power of a power rule
Using the power of a power rule \((a^m)^n = a^{m \cdot n}\), we simplify both the numerator and the denominator:
- For the numerator:
\[ (p^{-2x})^{6} = p^{-12x} \]
\[ (q^{3y})^{6} = q^{18y} \]
So, the numerator becomes:
\[ p^{-12x}q^{18y} \]

- For the denominator:
\[ (p^{3})^{-4x} = p^{-12x} \]
\[ (q^{-1})^{-4x} = q^{4x} \]
So, the denominator becomes:
\[ p^{-12x}q^{4x} \]

### Step 3: Substitute back into the expression
Now we substitute the simplified numerator and denominator back into the expression:
\[ \frac{p^{-12x}q^{18y}}{p^{-12x}q^{4x}} \]

### Step 4: Simplify the fraction
Since we have the same base \(p\) in both the numerator and denominator, we can subtract the exponents:
\[ p^{-12x - (-12x)} = p^{0} = 1 \]
For the \(q\) terms, we also subtract the exponents:
\[ q^{18y - 4x} \]

### Step 5: Final result
Thus, the expression simplifies to:
\[ q^{18y - 4x} \]