The solution set of the expression `1/((1 + p)^n) = 1` is :

Options

A. a.`[-1, 2]`
B. b.`(0,1)`
C. c.`{-2,0}`
D. d.`{-1,1}`

Answer

To solve the expression \( \frac{1}{(1 + p)^n} = 1 \), we will follow these steps:

### Step 1: Understand the equation
We start with the equation:
\[ \frac{1}{(1 + p)^n} = 1 \]
This means that the left-hand side must equal the right-hand side.

**Hint:** Recognize that for a fraction to equal 1, the numerator must equal the denominator.

### Step 2: Cross-multiply
To eliminate the fraction, we can cross-multiply:
\[ 1 = (1 + p)^n \]

**Hint:** Remember that multiplying both sides by the same non-zero quantity does not change the equality.

### Step 3: Analyze the equation
Now we have:
\[ (1 + p)^n = 1 \]
The expression \( (1 + p)^n \) equals 1 when \( 1 + p = 1 \) or when \( n = 0 \) (since any number to the power of 0 is 1).

**Hint:** Consider the properties of exponents and when they yield a result of 1.

### Step 4: Solve for \( p \)
From \( 1 + p = 1 \):
\[ p = 0 \]

**Hint:** Isolate \( p \) to find its value.

### Step 5: Consider the case when \( n = 0 \)
If \( n = 0 \), then \( (1 + p)^0 = 1 \) is true for any value of \( p \).

**Hint:** Think about the implications of exponents, especially with zero.

### Conclusion
The solution set of the expression \( \frac{1}{(1 + p)^n} = 1 \) is:
- \( p = 0 \) when \( n \neq 0 \)
- Any value of \( p \) when \( n = 0 \)

Thus, the complete solution set can be expressed as:
- If \( n = 0 \): all real numbers \( p \)
- If \( n \neq 0 \): \( p = 0 \)