If ` A = 1 + 2^(p) and B = 1 + 2^(-p)` , then what is the value of B ?

Options

A. `((A + 1))/( ( A - 1))`
B. ` ((A + 2))/( ( A + 1))`
C. `(A)/((A - 1))`
D. `((A - 2))/(( A + 1))`

Answer

To find the value of \( B \) given that \( A = 1 + 2^p \) and \( B = 1 + 2^{-p} \), we can follow these steps:

### Step 1: Express \( A \) in terms of \( 2^p \)
We know that:
\[ A = 1 + 2^p \]
From this, we can isolate \( 2^p \):
\[ 2^p = A - 1 \]

**Hint:** Rearranging the equation helps to express one variable in terms of another.

### Step 2: Substitute \( 2^{-p} \) in terms of \( A \)
Since \( 2^{-p} \) is the reciprocal of \( 2^p \), we can write:
\[ 2^{-p} = \frac{1}{2^p} = \frac{1}{A - 1} \]

**Hint:** Remember that the negative exponent indicates a reciprocal.

### Step 3: Substitute \( 2^{-p} \) into the expression for \( B \)
Now we can substitute \( 2^{-p} \) into the equation for \( B \):
\[ B = 1 + 2^{-p} = 1 + \frac{1}{A - 1} \]

**Hint:** Substitute the expression you found for \( 2^{-p} \) directly into the equation for \( B \).

### Step 4: Simplify the expression for \( B \)
To simplify \( B \), we need to combine the terms:
\[ B = 1 + \frac{1}{A - 1} = \frac{(A - 1) + 1}{A - 1} = \frac{A}{A - 1} \]

**Hint:** When adding fractions, find a common denominator.

### Conclusion
Thus, the value of \( B \) is:
\[ B = \frac{A}{A - 1} \]

### Final Answer
The final answer for \( B \) is \( \frac{A}{A - 1} \).

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