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

Options

A. a) `7`
B. b) `-7`
C. c) `(1)/(7)`
D. 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
First, we need to simplify the expression inside the brackets:
\[ 3 - 4(3 - 4)^{-1} \]
Calculate \(3 - 4\):
\[ 3 - 4 = -1 \]
Now substitute this back into the expression:
\[ 3 - 4(-1)^{-1} \]

### Step 2: Evaluate the inverse
Now, we need to evaluate \((-1)^{-1}\):
\[ (-1)^{-1} = -1 \]
So, we can rewrite the expression as:
\[ 3 - 4(-1) \]

### Step 3: Multiply and simplify
Now, multiply:
\[ 4 \times -1 = -4 \]
Thus, the expression simplifies to:
\[ 3 - (-4) = 3 + 4 = 7 \]

### Step 4: Apply the outer exponent
Now we have:
\[ [7]^{-1} \]
This means we take the reciprocal of 7:
\[ \frac{1}{7} \]

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

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