Find the value of `[1-2(1-2)^(-1)]^(-1)`

Options

A. `1/3`
B. `-1/3`
C. 1
D. `1/2`

Answer

To find the value of the expression \([1 - 2(1 - 2)^{-1}]^{-1}\), we will follow these steps:

### Step 1: Simplify the inner expression
Start with the expression inside the brackets:
\[ 1 - 2(1 - 2)^{-1} \]
First, calculate \(1 - 2\):
\[ 1 - 2 = -1 \]
Now, substitute this back into the expression:
\[ 1 - 2(-1)^{-1} \]

### Step 2: Calculate the inverse
Next, calculate \((-1)^{-1}\):
\[ (-1)^{-1} = -1 \]
Now substitute this back into the expression:
\[ 1 - 2(-1) \]

### Step 3: Simplify the expression
Now, simplify \(2(-1)\):
\[ 2(-1) = -2 \]
So now the expression becomes:
\[ 1 - (-2) = 1 + 2 = 3 \]

### Step 4: Take the inverse of the entire expression
Now we need to take the inverse of the entire expression:
\[ [3]^{-1} = \frac{1}{3} \]

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

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