Which of the following is equivalent to `(1/3)^(-4)(1/9)^(-3)(1/27)^(-2)` ?

Options

A. `(1/3)^(-8)`
B. `(1/3)^(-9)`
C. `(1/3)^(-16)`
D. `(1/3)^(-18)`

Answer

To solve the expression \((1/3)^{-4} \cdot (1/9)^{-3} \cdot (1/27)^{-2}\), we can follow these steps:

### Step 1: Rewrite the bases
First, we can express \(1/9\) and \(1/27\) in terms of \(1/3\):
- \(1/9 = (1/3)^2\)
- \(1/27 = (1/3)^3\)

So, we can rewrite the expression as:
\[ (1/3)^{-4} \cdot ((1/3)^2)^{-3} \cdot ((1/3)^3)^{-2} \]

### Step 2: Apply the power of a power rule
Next, we apply the power of a power rule, which states that \((a^m)^n = a^{m \cdot n}\):
- \(((1/3)^2)^{-3} = (1/3)^{-6}\)
- \(((1/3)^3)^{-2} = (1/3)^{-6}\)

Now, we can rewrite the expression as:
\[ (1/3)^{-4} \cdot (1/3)^{-6} \cdot (1/3)^{-6} \]

### Step 3: Combine the exponents
Since the bases are the same, we can add the exponents:
\[ (1/3)^{-4 + (-6) + (-6)} = (1/3)^{-4 - 6 - 6} \]

Calculating the exponent:
\[ -4 - 6 - 6 = -16 \]

### Step 4: Final expression
Thus, we have:
\[ (1/3)^{-16} \]

### Step 5: Rewrite in positive exponent form
Using the property \(a^{-n} = \frac{1}{a^n}\), we can rewrite this as:
\[ \frac{1}{(1/3)^{16}} = (3)^{16} \]

### Final Answer
The expression \((1/3)^{-4} \cdot (1/9)^{-3} \cdot (1/27)^{-2}\) is equivalent to \((3)^{16}\).

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