Solve or simplify the following problem, using the properties of roots: `(1/81)^(-1/4)`

Options

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

Answer

To solve the expression \((1/81)^{-1/4}\), we can follow these steps:

### Step 1: Apply the property of exponents
We know that \(a^{-n} = \frac{1}{a^n}\). Therefore, we can rewrite the expression as:
\[ (1/81)^{-1/4} = \frac{1}{(1/81)^{1/4}} \]

### Step 2: Simplify the denominator
Next, we can simplify \((1/81)^{1/4}\). This can be rewritten using the property of exponents:
\[ (1/81)^{1/4} = \frac{1^{1/4}}{81^{1/4}} = \frac{1}{81^{1/4}} \]

### Step 3: Rewrite \(81\) in terms of its prime factor
We know that \(81 = 3^4\). Therefore, we can substitute this into our expression:
\[ 81^{1/4} = (3^4)^{1/4} \]

### Step 4: Apply the property of exponents again
Using the property \((a^m)^n = a^{m \cdot n}\), we can simplify:
\[ (3^4)^{1/4} = 3^{4 \cdot (1/4)} = 3^1 = 3 \]

### Step 5: Substitute back into the expression
Now we can substitute back into our expression:
\[ (1/81)^{-1/4} = \frac{1}{(1/81)^{1/4}} = \frac{1}{\frac{1}{3}} = 3 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{3} \]

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