The expression `x (x ^(3) y ^(2)) ^(-4)` is equivalent to which of the following

Options

A. ` (1)/( y ^(2))`
B. ` (1)/(x ^(4))`
C. `(1)/( x ^(11) y ^(8)) `
D. `(1)/(x ^(16)y ^(8))`

Answer

To solve the expression \( x (x^{3} y^{2})^{-4} \), we will follow these steps:

### Step 1: Rewrite the expression
Start with the original expression:
\[ x (x^{3} y^{2})^{-4} \]

### Step 2: Apply the power of a product rule
Using the property that \( (a \cdot b)^{-n} = a^{-n} \cdot b^{-n} \), we can rewrite the expression inside the parentheses:
\[ = x \cdot (x^{3})^{-4} \cdot (y^{2})^{-4} \]

### Step 3: Simplify the powers
Now, apply the power rule \( a^{m \cdot n} = a^{m \cdot n} \):
\[ = x \cdot x^{-12} \cdot y^{-8} \]
Here, \( (x^{3})^{-4} = x^{-12} \) and \( (y^{2})^{-4} = y^{-8} \).

### Step 4: Combine the x terms
Now, combine \( x \) and \( x^{-12} \):
\[ = x^{1} \cdot x^{-12} \cdot y^{-8} \]
Using the property \( x^{m} \cdot x^{n} = x^{m+n} \):
\[ = x^{-11} \cdot y^{-8} \]

### Step 5: Rewrite in positive exponents
To express the result with positive exponents, we can rewrite the expression:
\[ = \frac{1}{x^{11} \cdot y^{8}} \]

### Final Result
Thus, the expression \( x (x^{3} y^{2})^{-4} \) is equivalent to:
\[ \frac{1}{x^{11} y^{8}} \]

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