The value of `(512)^(-(3)/(9))` is

Options

A. `1//4`
B. 8
C. `(1)/(8)`
D. `-(1)/(8)`

Answer

To solve the expression \( (512)^{-\frac{3}{9}} \), we will follow these steps:

### Step 1: Simplify the exponent
The exponent \(-\frac{3}{9}\) can be simplified.

\[ -\frac{3}{9} = -\frac{1}{3} \]

So, we can rewrite the expression as:

\[ (512)^{-\frac{1}{3}} \]

### Step 2: Rewrite the base
Next, we need to express \(512\) as a power of a smaller base. We know that:

\[ 512 = 2^9 \]

### Step 3: Substitute the base into the expression
Now we can substitute \(512\) in our expression:

\[ (2^9)^{-\frac{1}{3}} \]

### Step 4: Apply the power of a power rule
Using the power of a power rule, which states that \((a^m)^n = a^{m \cdot n}\), we can simplify:

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

### Step 5: Simplify the expression
Now we simplify \(2^{-3}\):

\[ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \]

### Final Answer
Thus, the value of \((512)^{-\frac{3}{9}}\) is:

\[ \frac{1}{8} \]

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