The value of `{ (8)^((-2)/(3)) }^(2)` is

Options

A. `1/4`
B. `1/8`
C. `16`
D. `1/16`

Answer

To solve the expression \({ (8)^{(-2/3)} }^{2}\), we will follow these steps:

### Step 1: Apply the Power of a Power Property
Using the property of exponents that states \((a^m)^n = a^{m \cdot n}\), we can rewrite the expression:
\[ (8)^{(-2/3) \cdot 2} \]

### Step 2: Simplify the Exponent
Now, we simplify the exponent:
\[ (-2/3) \cdot 2 = -4/3 \]
So, we have:
\[ 8^{-4/3} \]

### Step 3: Rewrite 8 as a Power of 2
Next, we can express 8 as a power of 2:
\[ 8 = 2^3 \]
Thus, we can rewrite the expression:
\[ (2^3)^{-4/3} \]

### Step 4: Apply the Power of a Power Property Again
Using the same property of exponents again:
\[ 2^{3 \cdot (-4/3)} = 2^{-4} \]

### Step 5: Simplify the Expression
Now we simplify \(2^{-4}\):
\[ 2^{-4} = \frac{1}{2^4} \]

### Step 6: Calculate \(2^4\)
Calculating \(2^4\):
\[ 2^4 = 16 \]
So, we have:
\[ 2^{-4} = \frac{1}{16} \]

### Final Answer
Thus, the value of \({ (8)^{(-2/3)} }^{2}\) is:
\[ \frac{1}{16} \]

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