`[(3-4(3-4)^(-1)]^(-1)` is equal to :

Options

A. 7
B. -7
C. `(1)/(7)`
D. `-(1)/(7)`

Answer

To solve the expression \([(3 - 4(3 - 4)^{-1})]^{-1}\), we will follow these steps:

### Step 1: Simplify the inner expression
Start with the expression inside the brackets:
\[ 3 - 4(3 - 4)^{-1} \]
First, we need to calculate \((3 - 4)^{-1}\):
\[ 3 - 4 = -1 \quad \Rightarrow \quad (3 - 4)^{-1} = (-1)^{-1} = -1 \]
Now substitute this back into the expression:
\[ 3 - 4(-1) \]

### Step 2: Multiply and simplify
Now perform the multiplication:
\[ 3 - 4 \times -1 = 3 + 4 = 7 \]

### Step 3: Apply the outer exponent
Now we have:
\[ [7]^{-1} \]
This means we take the reciprocal of 7:
\[ \frac{1}{7} \]

### Final Answer
Thus, the expression \([(3 - 4(3 - 4)^{-1})]^{-1}\) simplifies to:
\[ \frac{1}{7} \]