The value of `root 4((81)^(-2))` is

Options

A. `1/9`
B. `1/3`
C. 9
D. `-1/9`

Explanation

To solve the expression \( \sqrt[4]{(81)^{-2}} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt[4]{(81)^{-2}} \]

### Step 2: Apply the exponent property
Using the property of exponents that states \( a^{-n} = \frac{1}{a^n} \), we can rewrite \( (81)^{-2} \) as:
\[ \sqrt[4]{\frac{1}{(81)^2}} \]

### Step 3: Simplify the expression
Now we can simplify the expression further:
\[ \sqrt[4]{\frac{1}{(81)^2}} = \frac{1}{\sqrt[4]{(81)^2}} \]

### Step 4: Calculate \( (81)^2 \)
Calculating \( (81)^2 \):
\[ (81)^2 = 6561 \]
So, we have:
\[ \frac{1}{\sqrt[4]{6561}} \]

### Step 5: Find \( \sqrt[4]{6561} \)
Next, we need to find \( \sqrt[4]{6561} \). We can express 6561 as \( 81^{2} \) or \( (3^4)^{2} = 3^8 \). Therefore:
\[ \sqrt[4]{6561} = \sqrt[4]{3^8} = 3^{8/4} = 3^2 = 9 \]

### Step 6: Final calculation
Now, substituting back, we get:
\[ \frac{1}{\sqrt[4]{6561}} = \frac{1}{9} \]

### Final Answer
Thus, the value of \( \sqrt[4]{(81)^{-2}} \) is:
\[ \frac{1}{9} \]

---