If p is a number between 0 and 1, which one of the following is true?

Options

A. `pgtsqrt(p)`
B. `1/pltsqrt(p)`
C. `plt1/p`
D. `p^(3)gtp^(2)`

Answer

To solve the problem, we need to analyze the statements given about the number \( p \) which lies between 0 and 1. Let's go through each option step by step.

### Step 1: Understand the properties of numbers between 0 and 1
When \( p \) is a number between 0 and 1, any power of \( p \) (i.e., \( p^a \) where \( a > 1 \)) will be less than \( p \). This is because multiplying a fraction (which is less than 1) by itself results in a smaller number.

### Step 2: Analyze each option

#### Option A: \( \sqrt{p} > 1 \)
- Since \( p \) is between 0 and 1, \( \sqrt{p} \) will also be less than 1. Therefore, this statement is **false**.

#### Option B: \( p \sqrt{p} > 1 \)
- Rearranging gives \( p^{3/2} > 1 \). Since \( p \) is less than 1, \( p^{3/2} \) will also be less than 1. Therefore, this statement is **false**.

#### Option C: \( p^2 < 1 \)
- Since \( p \) is between 0 and 1, squaring \( p \) will yield a number that is still less than 1. Therefore, this statement is **true**.

#### Option D: \( p^3 > p^2 \)
- Since \( p \) is between 0 and 1, \( p^3 \) will be less than \( p^2 \). Therefore, this statement is **false**.

### Conclusion
The only true statement is Option C: \( p^2 < 1 \).

### Final Answer
The correct answer is **Option C: \( p^2 < 1 \)**.

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