If `9^(-2)=((1)/(3))^(x)`, what is the value of x?

Options

A. `1`
B. `2`
C. `4`
D. `6`

Answer

To solve the equation \( 9^{-2} = \left(\frac{1}{3}\right)^{x} \), we will follow these steps:

### Step 1: Rewrite \( 9^{-2} \)
We know that \( 9 \) can be expressed as \( 3^2 \). Therefore, we can rewrite \( 9^{-2} \) as follows:
\[ 9^{-2} = (3^2)^{-2} \]

### Step 2: Apply the power of a power property
Using the property of exponents that states \( (a^m)^n = a^{m \cdot n} \), we can simplify:
\[ (3^2)^{-2} = 3^{-4} \]

### Step 3: Rewrite \( 3^{-4} \) in terms of \( \frac{1}{3} \)
Now, we can express \( 3^{-4} \) as:
\[ 3^{-4} = \frac{1}{3^4} \]

### Step 4: Calculate \( 3^4 \)
Calculating \( 3^4 \):
\[ 3^4 = 81 \]
Thus, we have:
\[ 3^{-4} = \frac{1}{81} \]

### Step 5: Set the equation
Now we can rewrite the original equation:
\[ \frac{1}{81} = \left(\frac{1}{3}\right)^{x} \]

### Step 6: Rewrite \( \frac{1}{81} \) in terms of \( \frac{1}{3} \)
We can express \( 81 \) as \( 3^4 \):
\[ \frac{1}{81} = \frac{1}{3^4} = \left(\frac{1}{3}\right)^{4} \]

### Step 7: Set the exponents equal
Now we have:
\[ \left(\frac{1}{3}\right)^{4} = \left(\frac{1}{3}\right)^{x} \]
Since the bases are the same, we can set the exponents equal to each other:
\[ 4 = x \]

### Conclusion
Thus, the value of \( x \) is:
\[ \boxed{4} \]