Which expression is equivalent to `(9x^(2)y^(6))^(-(1)/(2))`?

Options

A. `(1)/(3xy^(3))`
B. `3xy^(3)`
C. `(3)/(xy^(3))`
D. `(xy^(3))/(3)`

Answer

To solve the expression \((9x^{2}y^{6})^{-\frac{1}{2}}\), we will follow these steps:

### Step 1: Apply the Power of a Product Rule
We can use the property of exponents that states \((a \cdot b \cdot c)^{m} = a^{m} \cdot b^{m} \cdot c^{m}\). Thus, we can rewrite the expression as:

\[ (9x^{2}y^{6})^{-\frac{1}{2}} = 9^{-\frac{1}{2}} \cdot (x^{2})^{-\frac{1}{2}} \cdot (y^{6})^{-\frac{1}{2}} \]

### Step 2: Simplify Each Component
Now, we simplify each part of the expression:

1. **For \(9^{-\frac{1}{2}}\)**:
\[ 9 = 3^{2} \implies 9^{-\frac{1}{2}} = (3^{2})^{-\frac{1}{2}} = 3^{2 \cdot -\frac{1}{2}} = 3^{-1} = \frac{1}{3} \]

2. **For \((x^{2})^{-\frac{1}{2}}\)**:
\[ (x^{2})^{-\frac{1}{2}} = x^{2 \cdot -\frac{1}{2}} = x^{-1} = \frac{1}{x} \]

3. **For \((y^{6})^{-\frac{1}{2}}\)**:
\[ (y^{6})^{-\frac{1}{2}} = y^{6 \cdot -\frac{1}{2}} = y^{-3} = \frac{1}{y^{3}} \]

### Step 3: Combine the Results
Now we combine all the simplified components:

\[ (9x^{2}y^{6})^{-\frac{1}{2}} = \frac{1}{3} \cdot \frac{1}{x} \cdot \frac{1}{y^{3}} = \frac{1}{3xy^{3}} \]

### Final Answer
Thus, the expression \((9x^{2}y^{6})^{-\frac{1}{2}}\) is equivalent to:

\[ \frac{1}{3xy^{3}} \]