`(1)/(x^(3)),(1)/(x^(2)), (1)/(x),x^(2),x^(3)`
If `-1ltxlt0`, what is the median of the five numbers in the list above?

Options

A. `(1)/(x^(2))`
B. `(1)/(x^(2))`
C. `(1)/(x)`
D. `x^(3)`

Answer

To find the median of the given numbers \(\frac{1}{x^3}, \frac{1}{x^2}, \frac{1}{x}, x^2, x^3\) when \(-1 < x < 0\), we will follow these steps:

### Step 1: Understand the range of \(x\)
Given that \(x\) is in the range \(-1 < x < 0\), we know that \(x\) is negative.

### Step 2: Analyze each expression
We will evaluate each expression in the list:
- \(\frac{1}{x^3}\)
- \(\frac{1}{x^2}\)
- \(\frac{1}{x}\)
- \(x^2\)
- \(x^3\)

### Step 3: Determine the signs and order of the expressions
1. **For \(\frac{1}{x^3}\)**: Since \(x\) is negative, \(x^3\) is also negative, making \(\frac{1}{x^3}\) positive.
2. **For \(\frac{1}{x^2}\)**: \(x^2\) is positive (since squaring a negative number yields a positive result), hence \(\frac{1}{x^2}\) is also positive.
3. **For \(\frac{1}{x}\)**: Since \(x\) is negative, \(\frac{1}{x}\) is negative.
4. **For \(x^2\)**: As mentioned, \(x^2\) is positive.
5. **For \(x^3\)**: \(x^3\) is negative.

### Step 4: Order the expressions from least to greatest
From the analysis:
- Negative values: \(x^3\) and \(\frac{1}{x}\)
- Positive values: \(x^2\), \(\frac{1}{x^2}\), and \(\frac{1}{x^3}\)

Now we can order them:
1. \(x^3\) (most negative)
2. \(\frac{1}{x}\) (less negative)
3. \(x^2\) (first positive)
4. \(\frac{1}{x^2}\) (greater positive)
5. \(\frac{1}{x^3}\) (largest positive)

So the ordered list is:
\[ x^3, \frac{1}{x}, x^2, \frac{1}{x^2}, \frac{1}{x^3} \]

### Step 5: Find the median
The median is the middle value in an ordered list. Since we have 5 numbers, the median is the 3rd number.

From our ordered list:
- 1st: \(x^3\)
- 2nd: \(\frac{1}{x}\)
- 3rd: \(x^2\) (this is the median)

### Final Answer
The median of the five numbers is:
\[ \boxed{x^2} \]