x and y are real numbers satisfying the conditions `2 lt x lt 3` and `-8 lt y lt -7`. Which of the following expressions will have the least value ?

Options

A. `x^(2)y`
B. `xy^(2)`
C. `5xy`
D. None of these

Answer

To solve the problem step by step, we need to evaluate the expressions given the constraints on \( x \) and \( y \).

### Step 1: Understand the constraints
We have the following constraints:
- \( 2 < x < 3 \)
- \( -8 < y < -7 \)

### Step 2: Select values for \( x \) and \( y \)
To find the least value of the expressions, we can choose values for \( x \) and \( y \) within the given ranges. A reasonable choice is:
- Let \( x = 2.5 \) (which is between 2 and 3)
- Let \( y = -7.5 \) (which is between -8 and -7)

### Step 3: Evaluate each expression
Now we will evaluate each of the expressions given in the options.

#### Option A: \( x^2 y \)
\[ x^2 y = (2.5)^2 \cdot (-7.5) = 6.25 \cdot (-7.5) = -46.875 \]

#### Option B: \( x y^2 \)
\[ x y^2 = 2.5 \cdot (-7.5)^2 = 2.5 \cdot 56.25 = 140.625 \]

#### Option C: \( 5xy \)
\[ 5xy = 5 \cdot 2.5 \cdot (-7.5) = 12.5 \cdot (-7.5) = -93.75 \]

### Step 4: Compare the results
Now we have the values for each expression:
- Option A: \( -46.875 \)
- Option B: \( 140.625 \)
- Option C: \( -93.75 \)

### Step 5: Identify the least value
Among the values calculated:
- The least value is \( -93.75 \) from Option C.

### Conclusion
Thus, the expression that has the least value is:
\[ \text{Option C: } 5xy \]