`(16) ^( - 3//4) + 2 ^(-3) + (8) ^( -2//3)` is equivalent to -

Options

A. `1/2`
B. `3/2`
C. `9/2`
D. `5/4`

Answer

To solve the expression \( (16)^{-3/4} + 2^{-3} + (8)^{-2/3} \), we can follow these steps:

### Step 1: Rewrite the bases in terms of powers of 2
First, we know that:
- \( 16 = 2^4 \)
- \( 8 = 2^3 \)

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

### Step 2: Apply the power of a power rule
Using the power of a power rule \( (a^m)^n = a^{m \cdot n} \), we simplify each term:
- For \( (2^4)^{-3/4} \):
\[ (2^4)^{-3/4} = 2^{4 \cdot (-3/4)} = 2^{-3} \]
- For \( (2^3)^{-2/3} \):
\[ (2^3)^{-2/3} = 2^{3 \cdot (-2/3)} = 2^{-2} \]

Now, substituting back into the expression gives:
\[ 2^{-3} + 2^{-3} + 2^{-2} \]

### Step 3: Combine like terms
Now we can combine the terms:
\[ 2^{-3} + 2^{-3} = 2 \cdot 2^{-3} = 2^{-3 + 1} = 2^{-2} \]
So the expression simplifies to:
\[ 2^{-2} + 2^{-2} = 2 \cdot 2^{-2} = 2^{-2 + 1} = 2^{-1} \]

### Step 4: Calculate the final value
Now we know that:
\[ 2^{-1} = \frac{1}{2} \]

Thus, the final answer is:
\[ \frac{1}{2} \]

### Final Answer
The expression \( (16)^{-3/4} + 2^{-3} + (8)^{-2/3} \) is equivalent to \( \frac{1}{2} \).

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